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WORKS  OF  PROFESSOR  MANSFIELD  MERR1MAN, 


Published  by  JOHN  WILEY  &  SONS,  53  E.  Tenth 
Street,  New  York. 


A  TREATISE  ON  HYDRAULICS. 

Designed  as  a  Text-book  for  Technical  Schools  and  for  the 
use  of  Engineers.  By  Professor  Mansfield  Merriman,  Lehigh 
University.  Fourth  edition,  revised 8vo,  cloth,  $3  50 

"  As  a  whole  this  book  is  the  most  valuable  addition  to  the  literature 
of  hydraulic  science  which  has  yet  appeared  in  America,  and  we  do  not 
know  of  any  of  equal  value  anywhere  else." — Railroad  Gazette. 

"  With  a  tolerably  complete  knowledge  of  what  has  been  written  on 
Hydraulics  in  England,  France,  Germany,  United  States,  and  to  some 
extent  Italy,  I  have  no  hesitation  in  saying  that  I  hold  this  book  to  be 
the  best  treatise  for  students,  young  or  old,  yet  written.  It  better  pre- 
sents the  primary  essentials  of  the  art.1' — From  CLEMENS  HERSCHEL, 
Hydraulic  Engineer  of  the  Holyoke  Water  Power  Company. 

A  TEXT-BOOK  OF  THE  METHOD  OF  LEAST  SQUARES. 
By  Mansfield  Merriman.  C.E.,  Ph.D.,  Professor  of  Civil 
Engineering  in  Lehigh  University. 

Sixth  edition,  enlarged 8vo,  cloth,  2  00 

This  work  treats  of  the  law  of  probability  of  error,  the  ad- 
justment and  discussion  of  observations  arising  in  surveying, 
geodesy,  astronomy  and  physics,  and  the  methods  of  compar- 
ing their  degrees  of  precision.  This  edition  contains  answers 
to  many  of  the  problems  in  the  text. 

"  This  is  a  very  useful  and  much  needed  text-book.1' — Science. 

"  Even  the  casual  reader  cannot  fail  to  be  struck  with  the  value 
which  such  a  book  must  possess  to  the  working  engineer.  It  abounds 
in  illustrations  and  problems  drawn  directly  from  surveying,  geodesy, 
and  engineering." — Engineering  News. 

THE  MECHANICS  OF  MATERIALS  AND  OF  BEAMS, 
COLUMNS,  AND  SHAFTS. 

A  Text-Book  for  classes  in  engineering.  By  Professor  Mans- 
field Merriman,  Lehigh  University,  South  Bethlehem,  Pa. 
Fourth  edition,  re  vised  and  enlarged.  8vo,  cloth,  interleaved,  3  50 

"  We  cannot  commend  the  book  too  highly  to  the  consideration  of  alt 
Professors  of  Applied  Mechanics  and  Engineering  and  Technical  Schools 
and  Colleges,  and  we  think  a  general  introduction  of  the  work  will 
mark  an  advance  in  the  rationale  of  technical  instruction.1' — Ameri- 
can Engineer. 

"  The  mathematical  deductions  of  the  laws  of  strength  and  stiffness 
of  beams,  supported,  fixed,  and  continuous,  under  compression, tension, 
and  torsion,  and  of  columns,  are  elegant  and  complete.  As  in  previous 
books  by  the  same  author,  plenty  of  practical,  original,  and  modern  ex- 
amples are  introduced  as  problems." — Proceedings  Engineers'  Club  of 
Philadelphia. 


A  TEXT-BOOK  ON  ROOFS  AND  BRIDGES. 

Being  the  course  of  instruction  given  by  the  author  to  the 
students  of  civil  engineering  in  Lehigh  University. 
To  be  completed  in  four  parts. 

PART!.    STRESSES  IN  SIMPLE  TRUSSES.    By  Professor 
Mansfield  Menlman.     Third  edition.     8vo,  cloth $2  50 

"  The  author  gives  the  most  modern  practice  in  determining  the 
stresses  due  to  moving  loads,  taking  actual  typical  locomotive  wheel 
loads,  and  reproduces  the  Phoenix  Bridge  Co/s  diagram  for  tabulating 
wheel  movements.  The  whole  treatment  is  concise  and  very  clear  and 
elegant." — Railroad  Gazette. 

PART  II.    GRAPHIC  STATICS.     By  Professors  Mansfield 
Merriman  and  Henry  S.  Jacoby.     Second  edition.  8vo,  cloth,  2  50 

"  The  plan  of  this  book  is  simple  and  easily  understood;  and  as  the 
treatment  of  all  problems  is  graphical,  mathematics  can  scarcely  be 
said  to  enter  into  its  composition.  Judging  from  our  own  correspond- 
ence, it  is  a  work  for  which  there  is  a  decided  demand  outside  of  tech- 
nical schools."— Engineering  News. 

PART  III.    BRIDGE  DESIGN.    In  Preparation. 

This  volume  is  intended  to  include  the  design  of  plate  girders, 
lattice  trusses,  and  pin-connected  bridges,  together  with  the 
proportioning  of  details,  the  whole  being  in  accordance  with 
-  the  best  modern  practice  and  especially  adapted  to  the  needs 
of  students. 

RETAINING  WALLS  AND  MASONRY  DAMS. 

By  Mansfield  Merriman,  C.E 8vo,  cloth,  $2  00 

This  Text-book  treats  of  earthwork  slopes,  of  computing  the 
thrust  of  earth  against  walls,  and  of  the  investigation  and  de- 
sign of  walls  and  dams. 

"  It  is  a  work  of  great  practical  value  to  the  army  of  irrigation  en- 
gineers in  the  West.1'— Irrigation  Age. 

"  It  is  altogether  an  admirable  text-book  on  a  somewhat  difficult 
subject.— Engineering  and  Mining  Journal." 

"  The  mathematical  work  is  simple  and  easily  followed.  Everything 
in  the  book  has  a  direct  practical  bearing.—  Engineering  News. 

AN  INTRODUCTION  TO  GEODETIC  SURVEYING. 

By  Mansfield  Merriman,  Ph.D.,  late  Acting  Assistant,  U.  S. 

Coast  and  Geodetic  Survey 8vo,  cloth,  $2  00 

This  book  has  three  divisions:  I.  Lectures  on  the  Figure  of 
the  Earth,  II.  The  Principles  of  the  Method  of  Least  Squares, 
III.  Methods  and  Computations  in  the  Field  Work  of  Tri- 
angulation; — the  whole  forming  an  elementary  course  of  In- 
struction in  Geodesy. 


A   TREATISE 


ON 


HYDRAULICS. 


BY 

MANSFIELD  MERRIMAN, 

PROFESSOR  OF  CIVIL  ENGINEERING IN -LEHIGH  UNIVERSITY. 


FOURTH  REVISED  EDITION. 
THIRD    THOUSAND. 


NEW  YORK: 

JOHN    WILEY    &    SONS, 

53   EAST   TENTH   STREET. 

1893. 


library 


Copyright,  1889, 

BY 
MANSFIELD  MERRIMAN. 


SIFT  OF 


u4AvW     4    VjJ- 


ENGINEERING  LIBRARY 


q 


DBUMMOND  &  Nmr, 

Electrotype™, 

1  to  7  Hagrie  Street. 

New  York. 


FKBBIS  BEOS 

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New  york> 


CONTENTS. 


PAGE 

Chapter  I,  INTRODUCTION i 

Art.  i.  Units  of  Measure     .         .         .        .        .        .        .        ..  •  i 

2.  Physical  Properties  of  Water  . 3 

3.  The  Weight  of  Water 5 

4.  Atmospheric  Pressure 7 

5.  Compressibility  of  Water        .........  9 

6.  The  Acceleration  of  Gravity    ........  10 

7.  Numerical  Computations          .......  12 

Chapter  II.  HYDROSTATICS .  14 

Art.  8.  Transmission  of  Pressures       .......  14 

9.  Head  and  Pressure 15 

10.  Normal  Pressure r8 

11.  Pressure  in  a  given  Direction ,20 

12.  Centre  of  Pressure  on  Rectangles            .....  22 

13.  General  Rule  for  Centre  of  Pressure        ..                  .         .         .24 
•    14.  Pressures  on  Opposite  Sides  of  a  Plane  .         .         .  27 

15.  Masonry  Dams         .  .         .         .         .         .         .         .28 

16.  Loss  of  Weight  in  Water          .......  30 

17.  Depth  of  Flotation 31 

18.  Stability  of  Flotation 33 

Chapter  III.  THEORETICAL  HYDRAULICS 36 

Art.   19.  Velocity  and  Discharge 36 

20.  Velocity  of  Flow  from  Orifices         ......  37 

21.  Orifices  whose  Plane  is  Horizontal  ...         .         .         .40 

22.  Rectangular  Vertical  Orifices 42 

23.  Triangular  Vertical  Orifice^ 44 

24.  Circular  Vertical  Orifices 45 

25.  Influence  of  Velocity  of  Approach 46 

26.  Flow  under  Pressure         ........  49 

27.  Pressure  Head  and  Velocity  Head  ......  52 

28.  Time  of  Emptying  a  Vessel 56 

29.  Flow  from  a  revolving  vessel 58 

iii 


863904 


IV  CONTENTS. 

PAGE 

Art.  30.  The  Path  of  a  Jet    . 61 

31.  The  Energy  of  a  Jet 64 

32.  The  Impulse  and  Reaction  of  a  Jet          .....  66 

33.  Absolute  and  Relative  Velocities     ......  68 

Chapter  IV.  FLOW  THROUGH  ORIFICZS 71 

Art.  34.  The  Standard  Orifice  • 71 

35.  The  Coefficient  of  Contraction          .         .         .         .  -73 

36.  The  Coefficient  of  Velocity       ...         .         .         .         .  74 

37.  The  Coefficient  of  Discharge   .......  76 

38.  Vertical  Circular  Orifices           .         ...         .         ..-.-,,.  -78 

39*.  Vertical  "Square  Orifices          "  »         .         .     ,    .         .         .         .  $o 

40.  Vertical  Rectangular  Orifices  .   »    .         .."'•.»".      •         •         •  82 

41°.  The  Miner's  Inch     .         .         .         .         .         .      r  .     ,    .         .  84 

42°.  Submerged  Orifices           .         .         .         .         .         .     "    1         .  86 

43°.  Suppression  of  the  Contraction        .         .         .         .         .         .  87 

44!  Orifices  with  Rounded  Edges  .         .         .       .".     ,    .    ',.   .         .  88 

45^  Measurement' of  Water  by  Orifices         .         .         .        ".        .  89 

46!  The  Energy  o°f  the 'Discharge  .         .    '     .'      .         .     "   '.         .  92 

47.  Discharge  under  a  Variable  Head    ..       .         ...       -,*•        .  ,94 

48.°  Emptying  and  Filling  a  "Canal  Lock      '  .  ''.'  »        ...         .'97 

Chapter  V.  FLOW  OVER  WEIRS  \      '.        .,'.,.        .     :..      .100 

,  Art.  49.  Description  of  a  Weir       »         .         .         .         .         .         .         .  100 

50.  The  Hook  Gauge     .         .         .        .         .         .         •        •         •  IO2 

51.  Formulas  for  the  Discharge     .         .         .        .         .         .         .  104 

52.  The  Velocity  of  Approach        .         .         .         .         .      „  .        ..  107 

53.  Weirs  with  End  Contractions  .         .         .         ...         .  109 

54.  Weirs  without  End  Contractions      .         .        .         .      >  .    •     .  112 

55.  FRANCIS' Formulas  .         .         .        .         .         .  114 

56.  Submerged  Weirs     .         .         .         .         .         .         .         .         .  1 16 

57.  Rounded  and  Wide  Crests       .         .         .         .         .         .         .  119 

58.  Waste  Weirs  and  Dams  .         ...         ,         .         .         .  121 

59.  The  Surface  Curve    .         .         .         .         .         .         ...  124 

60.  Triangular  Weirs     .        .         .        .         .      ...       .        .        .  126 

Chapter  VI.  FLOW^  THROUGH  TVBES.       .       .       .       ...       .128 

Arf.  6i%  The  Standard, Shorf  Tut^e        .        .         .        .        „        ..   -    .  128 

,    62.,  Conjcal  Converging  Tubes       .         .         .         .         ....  130 

,    63.  Nozzles  ,.       ,.       ..         .         ,         .         .         .         .         ..    :    .  132 

64..  Diverging  and,  Compound  Tubes     .         ...         .         .  137 

,    65. „  Inward  Projecting  Tubes         ....         .         .     -    .  141 

.    66.,  Effective  HeacJ  and  Lost,  Head         .         .         .         .         .,        .  142 

„    67..  Losses  in,  the  Stand,ard  Tube  ,         .         .    .     .        .        .         .  145 


CONTENTS.  V 

PAGE 

Art.  68.  Loss  due  to  Enlargement  of  Section 148 

69.  Loss  due  to  Contraction  of  Section 151 

70.  Piezometers 153 

71.  The  Venturi  Water  Meter .158 

Chapter  VII.  FLOW  IN  PIPES 162 

Art.  72.   Fundamental  Ideas  .         .         .         .         .         .         .         .  162 

73.  Loss  of  Head  at  Entrance         .......  164 

74.  Loss  of  Head  in  Friction 166 

75.  Other  Losses  of  Head 170 

76.  Formula  for  Velocity 173 

77.  Computation  of  Discharge       ...         .         .         .         .         .  175 

78.  Computation  of  Diameter         .         .         .         .         .         .         .  177 

79.  Short  Pipes      .         .         .         .         .         .         .         .         .         .  179 

80.  Long  Pipes .         .         .         .  181 

81.  Relative  Discharging  Capacities 182 

82.  A  Compound  Pipe   .........  184 

83.  Piezometer  Measurements       .......  186 

84.  The  Hydraulic  Gradient            .......  189 

85.  A  Pipe  with  Nozzle 193 

86.  House-service  Pipes 196 

87.  A  Water  Main .198 

88.  A  Main  with  Branches     ........  201 

89.  Pumping  through  Pipes 203 

90.  Leather  and  Rubber  Hose 207 

91.  LAMPE'S  Formula 208 

92.  Very  Small  Pipes *  210 

Chapter  VIII.  FLOW  IN  CONDUITS  AND  CANALS    ....  212 

Art.  93.  Definitions .         .         .  212 

94.  Formula  for  Mean  Velocity      .                  .         .         .         .         .  215 

95.  Circular  Conduits,  full  or  half  full .  2i3 

96.  Circular  Conduits,  partly  full  .......  221 

97.  Open  Rectangular  Conduits     .         .         .         .         .         .         .  224 

98.  Trapezoidal  Sections 227 

99.  Horseshoe  Conduits         . .231 

100.  LAMPE'S  Formula 232 

101.  KUTTER'S  Formula  .        .         .        .         .        .         .         .        .  233 

102.  Sewers 235 

103.  Ditches  and  Canals                    .    "  • 239 

104.  Losses  of  Head         .         .               •  .         .         .         .         .         .  242 

105.  The  Energy  of  the  Flow           .......  244 

Chapter  IX.  FLOW  IN  RIVERS     .        .       .       ...       .        .  247 

Art.  106.  Brooks  and  Rivers  .         .        .        .        ....        .  247 


VI  CONTENTS. 

PAC3 

Art.   107.  Velocities  in  a  Cross-section    .......  249 

108.  The  Transporting  Capacity  of  Currents  .         .         .         .251 

109.  The  Current  Meter          .         .  * 253 

no.  Floats 256 

in.  Other  Current  Indicators 258 

112.  Gauging  the  Flow    .........  260 

113.  Gauging  by  Surface  Velocities          ......  262 

114.  Gauging  by  Sub-surface  Velocities  ......  264 

115.  Comparison  of  Methods           . 266 

116.  Variations  in  Velocity  and  Discharge       .....  268 

117.  Non-uniform  Flow            ........  270 

1 1 8.  The  Surface  Curve 273 

119.  Backwater 277 

Chapter  X.  MEASUREMENT  OF  WATER  POWER       ....  283 

Art.   120.  Theoretic  and  Effective  Power         ......  283 

121.  Measurement  of  the  Water      .......  285 

122.  Measurement  of  the  Head        .......  2^8 

123.  Determination  of  Effective  Power 290 

124.  The  Friction  Brake,  or  Power  Dynamometer          .         .         .  292 

125.  Test  of  a  Small  Motor 295 

126.  The  Lowell  and  Holyoke  Tests       ......  298 

127.  Water  Power  of  Rivers  and  of  the  Tides         ....  302 

Chapter  XI.  DYNAMIC  PRESSURE  OF  FLOWING  WATER        .       .  304 

Art.  128.  Definitions  and  Principles 304 

129."  Experiments  on  Impulse  and  Reaction    .         .         .         .         .  307 

130.  Surfaces  at  Rest        .         .         .         .                  .         .         .         .  310 

131.  Curved  Pipes  and  Channels 313 

132.  Immersed  Bodies 316 

133.  Moving  Vanes 318 

134.  Work  derived  from  Moving  Vanes  ......  322 

135.  Revolving  Vanes 326 

136.  Work  derived  from  Revolving  Vanes 328 

137.  Revolving  Tubes 332 

Chapter  XII.  HYDRAULIC  MOTORS 335 

Art.  138.  Conditions  of  high  efficiency 335 

139.  Overshot  Wheels 337 

140.  Breast  Wheels 341 

141.  Undershot  Wheels 343 

142.  Horizontal  Impulse  Wheels      .......  345 

143.  Reaction  Wheels 349 

144.  Flow  through  Turbine  Wheels 353 


CONTENTS.  Vll 


Art.   145.  Theory  of  Turbines          ........  358 

146.  Other  Kinds  of  Motors     .                 .        .        .        .        .        .  362 

Chapter  XIII.  NAVAL  HYDROMECHANICS 364 

Art.  147.  General  Principles  .........  365 

148.  Frictional  Resistances 367 

149.  Work  required  in  Propulsion  .......  370 

150.  The  Jet  Propeller 371 

151.  Paddle  Wheels •  -373 

152.  The  Screw  Propeller 375 

153.  Action  of  the  Rudder 378 

154.  Tides  and  Waves «...  379 


TABLES. 


PAGE 

I.  Weight  of  Distilled  Water      . ,        .6 

II.  Atmospheric  Pressure    . .  8 

III.  Heads  and  Pressures 17 

IV.  Theoretic  Velocities        .                  40 

V.  Velocity  Heads       §        .         .        , 40 

VI.  Coefficients  for  Circular  Vertical  Orifices       .         .         .         . .       .  79 

VII.  Coefficients  for  Square  Vertical  Orifices          .         .         .         .         .81 

VIII.  Coefficients  for  Rectangular  Orifices,  I  foot  wide           ...  83 

IX.  Coefficients  for  Submerged  Orifices        ......  87 

X.  Coefficients  for  Contracted  Weirs  .......  no 

XI.  Coefficients  for  Suppressed  Weirs  .         .         .         .                  .         .  113 

XII.  Submerged  Weirs 117 

XIII.  Corrections  for  Wide  Crests   .         . 120 

XIV.  Coefficients  for  Conical  Tubes        .         ...         .         .         .131 

XV.  Vertical  Heights  of  Jets  from  Nozzles 133 

XVI.  Friction  Factors  for  Pipes 168 

XVII.  Coefficients  for  Circular  Conduits 218 

XVIII.  Cross-sections  in  Circular  Conduits 222 

XIX.  Coefficients  for  Rectangular  Conduits 227 

XX.  Coefficients  for  Sewers 238 

XXI.  Coefficients  for  Channels  in  Earth-          ......  239 

XXII.  Values  of  the  Backwater  Function 280 

XXIII.  Test  of  a  6-inch  Eureka  Turbine 296 

XXIV.  Results  of  Test  of  a  6-inch  Turbine 297 

XXV.  Test  of  an  8o-inch  Boyden  Turbine        ......  300 


Evolvi  varia  problemata.     In  scientiis  enim  ediscendis  prosunt  exempla 
magis  quam  praecepta.     Qua  de  causa  in  his  fusius  expatiatus  sum. 

NEWTON. 


HYDRAULICS. 


CHAPTER   I. 
INTRODUCTION. 

ARTICLE  i.  UNITS  OF  MEASURE. 

The  unit  of  linear  measure  universally  adopted  in  English 
and  American  hydraulic  literature  is  the  foot,  which  is  defined 
as  one-third  of  the  standard  yard.  For  some  minor  purposes, 
such  as  the  designation  of  the  diameters  of  orifices  and  pipes, 
the  inch  is  employed,  but  inches  should  always  be  reduced  to 
feet  for  use  in  hydraulic  formulas.  The  unit  of  superficial 
measure  is  usually  the  square  foot,  except  for  the  expression  of 
the  intensity  of  pressures,  when  the  square  inch  is  more  com- 
monly employed. 

The  units  of  volume  employed  in  measuring  water  are  the 
cubic  foot  and  the  gallon.  In  Great  Britain  the  Imperial  gal- 
lon is  used,  and  in  this  country  the  old  English  gallon,  the 
former  being  20  per  cent  larger  than  the  latter.  The  following 
are  the  relations  between  the  cubic  foot  and  the  two  gallons : 

i  cubic  foot  =6.232  Imp.  gallons  =  7.481  U.  S.  gallons; 
I  Imp. gallon  =  0.1605  cubic  feet  =  1.200  U.  S.  gallons; 
i  U.S. gallon  =  0.1337  cubic  feet  =0.8331  Imp.  gallons. 

In  this  book  the  word  gallon   will  always  mean  the  United 
States  gallon  of  231  cubic  inches,  unless  otherwise  stated. 


2  y  t  t/t  *^  t  INTRODUCTION.  [CHAP.  I. 

.\\  t'tjie\ui4it- of  fwejgfjt  js^the  avoirdupois  pound,  which  is  also 
the  unit  for  measuring  pressures.  The  intensity  of  pressure 
will  be  measured  in  pounds  per  square  foot  or  in  pounds  per 
square  inch,  as  may  be  most  convenient,  and  sometimes  in 
atmospheres  (Art.  4).  Gauges  for  recording  the  pressure  of 
water  are  usually  graduated  so  as  to  read  pounds  per  square 
inch. 

The  unit  of  time  used  in  all  hydraulic  formulas  is  the  second, 
although  in  numerical  problems  the  time  is  often  stat'ed  in 
minutes,  hours,  or  days.  Velocity  is  defined  as  the  space  passed 
over  by  a  body  in  one  second  under  the  condition  of  uniform 
motion,  so  that  velocities  are  to  be  always  expressed  in  feet  per 
second,  or  are  to  be  reduced  to  these  units  if  stated  in  miles  per 
hour  or  otherwise. 

The  unit  of  work,  or  energy,  is  the  foot-pound  ;  that  is,  one 
pound  lifted  through  a  vertical  distance  of  one  foot.  Energy 
is  potential  work,  or  the  work  which  can  be  done ;  for  example, 
a  moving  stream  of  water  has  the  ability  to  do  a  certain  amount 
of  work  by  virtue  of  its  weight  and  velocity,  and  this  is  called 
energy,  while  the  word  work  is  more  generally  used  for  that 
actually  done  by  a  motor  which  is  moved  by  the  water.  Power 
is  work,  or  energy,  done  or  existing  in  a  specified  time,  and  the 
unit  for  its  measure  is  the  horse-power,  which  is  550  foot-pounds 
per  second,  or  33  ooo  foot-pounds  per  minute. 

In  French  and  German  literature  the  metric  system  is  em- 
ployed ;  the  meter  and  centimeter  being  the  units  of  length,  and 
their  squares  the  units  of  superficial  measure.  The  units  of 
capacity  are  the  cubic  meter  and  the  liter,  that  of  weight  the 
kilogram,  and  that  of  time  the  second.  The  unit  of  work  is  the 
kilogram-meter,  and  one  horse-power  is  75  kilogram-meters, 
which  is  about  1.5  per  cent  less  than  that  as  defined  above. 
Students  should  be  prepared  to  rapidly  transform  metric  into 


ART.  2.]  PHYSICAL  PROPERTIES  OF    WATER.  3 

American  measures,  for  which  purpose  a  table  of  equivalents 
giving  logarithms  will  be  found  most  convenient.* 

The  motion  of  water  in  river  channels,  and  its  flow  through 
orifices  and  pipes,  is  produced  by  the  force  of  gravity.  This 
force  is  proportional  to  the  acceleration  of  the  velocity  of  a 
body  falling  freely  in  a  vacuum ;  that  is,  to  the  increase  in 
velocity  in  one  second.  The  acceleration  is  measured  in  feet 
per  second  per  second,  so  that  its  value  represents  the  number 
of  feet  per  second  which  have  been  gained  in  one  second  by  a 
falling  body. 

Problem  I.  How  many  pounds  per  square  inch  are 
equivalent  to  a  pressure  of  70  kilograms  per  square  centimeter? 

ARTICLE  2.  PHYSICAL  PROPERTIES  OF  WATER. 

At  ordinary  temperatures  pure  water  is  a  colorless  liquid 
which  possesses  perfect  fluidity ;  that  is,  its  particles  have  the 
capacity  of  moving  over 
each  other,  so  that  the 
slightest  disturbance  of 
equilibrium  causes  a  flow. 
It  is  a  consequence  of  this 
property  that  the  surface  FlG-  *• 

of  still  water  is  always  level ;  also,  if  several  vessels  or  tubes  be 
connected,  as  in  Fig.  i,  and  water  be  poured  into  one  of  them, 
it  rises  in  the  others  until,  when  equilibrium  ensues,  the  free 
surfaces  are  in  the  same  level  plane. 

The  free  surface  of  water  is  in  a  different  molecular  condi- 
tion from  the  other  portions,  its  particles  being  drawn  together 
by  stronger  attractive  forces,  so  as  to  form  what  may  be  called 
the  "  skin  of  the  water,"  upon  which  insects  walk.  The.  skin  is 
not  immediately  pierced  by  a  sharp  point  which  moves  slowly 

*  See  LANDRETH'S  Metrical  Tables  for  Engineers  (Philadelphia,  1883). 


4  INTRODUCTION.  [CHAP.  I. 

upward  toward  it,  but  a  slight  elevation  occurs,  and  this  prop- 
erty enables  precise  determinations  of  the  level  of  still  water 
to  be  made  by  means  of  the  hook  gauge  (Art.  50). 

At  about  32  degrees  Fahrenheit  a  great  alteration  in  the 
molecular  constitution  of  water  occurs,  and  ice  is  formed.  If  a 
quantity  of  water  be  kept  in  a  perfectly  quiet  condition,  it  is 
found  that  its  temperature  can  be  reduced  to  20°,  or  even  to 
15°,  Fahrenheit,  before  congelation  takes  place,  but  at  the 
moment  when  this  occurs  the  temperature  rises  to  32°.  The 
freezing-point  is  hence  not  constant,  but  the  melting-point  of 
ice  is  always  at  the  same  temperature  of  32°  Fahrenheit  or  o° 
Centigrade. 

Ice  being  lighter  than  water,  forms  as  a  rule  upon  its  sur- 
face ;  but  when  water  is  in  rapid  motion  a  variety  called  anchor 
ice  may  occur.  In  this  case  the  ice  is  formed  at  the  surface  in 
the  shape  of  small  needles,  which  are  quickly  carried  to  the 
lower  strata  by  the  agitation  due  to  the  motion ;  there  the 
needles  adhere  to  the  bed  of  the  stream,  sometimes  accumulat- 
ing to  an  extent  sufficient  to  raise  the  water  level  several  feet.* 
Anchor  ice  frequently  causes  obstructions  in  conduits  and 
orifices  which  lead  water  to  motors. 

Water  is  a  solvent  of  high  efficiency,  and  is  therefore  never 
found  pure  in  nature.  Descending  in  the  form  of  rain  it  ab- 
sorbs dust  and  gaseous  impurities  from  the  atmosphere ;  flow- 
ing over  the  surface  of  the  earth  it  absorbs  organic  and  mineral 
substances.  These  affect  its  weight  only  slightly  as  long  as  it 
remains  fresh,  but  when  it  has  reached  the  sea  and  become  salt 
its  weight  is  increased  more  than  two  per  cent.  The  flow  of 
water  through  orifices  and  pipes  is  only  in  a  very  slight  degree 
affected  by  the  impurities  held  in  solution. 

*  FRANCIS  in  Transactions  American  Society  Civil  Engineers,  1881,  vol.  x. 
p.  192. 


ART.  3.]  THE    WEIGHT  OF    WA  TER.  5 

The  capacity  of  water  for  heat,  the  latent  heat  evolved  when 
it  freezes,  and  that  absorbed  when  it  is  transformed  into  steam, 
need  not  be  considered  for  the  purposes  of  hydraulic  investiga- 
tions. Other  physical  properties,  such  as  its  variation  in  volume 
with  the  temperature,  its  compressibility,  and  its  capacity  for 
transmitting  pressures,  are  discussed  in  detail  in  the  following 
pages.  The  laws  which  govern  its  pressure,  flow,  and  energy 
under  various  circumstances  belong  to  the  science  of  Hydraulics, 
and  form  the  subject-matter  of  this  volume. 

Prob.  2.  What  horse-power  is  required  to  lift  16000 
pounds  of  water  per  minute  through  a  vertical  height  of  21 
feet?  Ans.  10.2. 

ARTICLE  3.  THE  WEIGHT  OF  WTATER. 

The  weight  of  water  per  unit  of  volume  depends  upon  the 
temperature  and  upon  its  degree  of  purity.  The  following 
approximate  values  are,  however,  those  generally  employed 
except  when  great  precision  is  required : 

I  cubic  foot  weighs     62.5      pounds; 
I  U.  S.  gallon  weighs    8.355  pounds. 

These  values  will  be  used  in  this  book,  unless  otherwise  stated, 
in  the  solution  of  the  examples  and  problems. 

The  weight  per  unit  of  volume  of  pure  distilled  water  is  the 
greatest  at  the  temperature  of  its  maximum  density,  39°-3 
Fahrenheit,  and  least  at  the  boiling-point.  For  ordinary  com- 
putations the  variation  in  weight  due  to  temperature  is  not 
considered,  but  in  tests  of  the  efficiency  of  hydraulic  motors 
and  of  pumps  it  should  be  regarded.  The  following  table  is 
hence  given,  which  contains  the  weights  of  one  cubic  foot  of 
pure  water  at  different  temperatures  as  deduced  by  SMITH 
from  the  experiments  of  ROSSETTL* 

*  HAMILTON  SMITH,  Jr.,  Hydraulics  :  The  Flow  of  Water  through  Orifices, 
over  Weirs,  and  through  open  Conduits  and  Pipes  (London  and  New  York, 
1886),  p.  14. 


INTRODUCTION.  [CHAP.  I. 

TABLE    I.     WEIGHT   OF   DISTILLED   WATER. 


Temperature 
(Fahrenheit). 

Pounds  per 
Cubic  Foot. 

Temperature 
(Fahrenheit). 

Pounds  per 
Cubic  Foot. 

Temperature 
(Fahrenheit). 

Pounds  per 
Cubic  Foot. 

32° 

62.42 

95 

62.06 

1  60 

6l.OI 

35 

62.42 

IOO 

62.OO 

165 

60.90 

39-3 

62.424 

105 

61.93 

170 

60.80 

45 

62.42 

no 

61.86 

175 

60.69 

50 

62.41 

H5 

61.79 

180 

60.59 

55 

62.39 

120 

61.72 

185 

60.48 

60 

62.37 

125 

61.64 

190 

60.36 

65 

62.34 

130 

61.55 

195 

60.25 

70 

62.30 

135 

61.47 

200 

60.14 

75 

62.26 

140 

61.39 

2O5 

60.02 

80 

62.22 

145 

61.30 

2IO 

59.89 

85 

62.17 

150 

61.20 

212 

59-84 

90 

62.12 

155 

6i.n 

Waters  of  rivers,  springs,  and  lakes  hold  in  suspension  and 
solution  inorganic  matters  which  cause  the  weight  per  unit  of 
volume  to  be  slightly  greater  than  for  pure  water.  River 
waters  are  usually  between  62.3  and  62.5  pounds  per  cubic  foot, 
depending  upon  the  amount  of  impurities  and  on  the  tempera- 
ture, while  the  water  of  some  mineral  springs  has  been  found 
to  be  as  high  as  62.7.  It  appears  that,  in  the  absence  of  specific 
information  regarding  a  particular  water,  the  wreight  62.5  pounds 
per  cubic  foot  is  a  fair  approximate  value  to  use.  It  also  has 
the  advantage  of  being  a  convenient  number  in  computations,  for 
62.5  pounds  is  1000  ounces,  or  if§-^  is  the  equivalent  of  62.5. 


In  the  metric  system  the  weight  of  a  cubic  meter  of  pure 
water  at  a  temperature  near  that  of  maximum  density  is  taken 
as  1000  kilograms,  which  is  the  average  unit-weight  used  in 
hydraulic  computations.  This  corresponds  to  62.426  pounds 
per  cubic  foot. 

Brackish  and  salt  waters  are  always  much  heavier  than  fresh 
water.  For  the  Gulf  of  Mexico  the  weight  per  cubic  foot  is 
about  63.9,  for  the  oceans  about  64.1,  while  for  the  Dead  Sea 
there  is  stated  the  value  73  pounds  per  cubic  foot.  The  weight 
of  ice  per  cubic  foot  varies  from  57.2  to  57.5  pounds. 


ART.  4.]  A  TMOSPHERIC  PRESSURE.  7 

Prob.  3.  How  many  pounds  of  water  in  a  cylindrical  box 
2  feet  in  diameter  and  2  feet  deep  ?  How  many  gallons  ?  How 
many  kilograms?  How  many  liters? 

Prob.  4.  In  a  certain  problem  regarding  the  horse-power 
required  to  lift  water,  the  computations  were  made  with  the 
mean  value  62.5  pounds  per  cubic  foot.  Supposing  that  the 
actual  weight  per  cubic  foot  was  62.35  pounds,  show  that  the 
error  thus  introduced  was  less  than  one-fourth  of  one  per  cent. 


ARTICLE  4.  ATMOSPHERIC  PRESSURE. 

The  pressure  of  the  atmosphere  is  measured  by  the  readings 
of  the  barometer.  This  instrument  is  a  tube  entirely  exhausted 
of  air,  which  is  inserted  into  a  vessel  containing  a  liquid.  The 
pressure  of  the  air  on  the  surface  of  the  liquid  causes  it  to  rise 
in  the  tube  until  it  attains  a  height  which  exactly  balances  the 
pressure  of  the  air.  Or  in  other  words,  the  weight  of  the  baro- 
metric column  is  equal  to  the  weight  of  a  column  of  air  of  the 
same  cross-section  as  that  of  the  tube,  both  columns  being 
measured  upward  from  the  surface  of  the  liquid  in  the  vessel. 
The  liquid  generally  employed  is  mercury,  and,  owing  to  its 
great  density,  the  height  of  the  column  required  to  balance  the 
atmospheric  pressure  is  only  about  30  inches,  whereas  a  water 
barometer  would  require  a  height  of  over  30  feet. 

The  atmosphere  exerts  its  pressure  with  varying  intensity, 
as  indicated  by  the  readings  of  the  mercury  barometer.  At 
and  near  the  sea  level  the  average  reading  is  30  inches,  and  as 
mercury  weighs  0.49  pounds  per  cubic  inch  at  common  tem- 
peratures, the  average  atmospheric  pressure  is  taken  to  'be 
30  X  0.49  or  14.7  pounds  per  square  inch. 

The  pressure  of  one  atmosphere  is  therefore  defined  to  be 
a  pressure  of  14.7  pounds  per  square  inch.  Then  a  pressure  of 
two  atmospheres  is  29.4  pounds  per  square  inch.  And  con- 


8 


INTRODUCTION. 


[CHAP.  I- 


versely,  a  pressure  of  one  pound  per  square  inch  may  be  expressed 
as  a  pressure  of  0.068  atmospheres. 

The  rise  of  water  in  a  vacuum  is  due  merely  to  the  pressure 
lof  the  atmosphere,  like  that  of  the  mercury  in  the  common 
barometer.  In  a  perfect  vacuum,  water  will  rise  to  a  height  of 
about  34  feet  under  the  mean  pressure  of  one  atmosphere,  for 
the  specific  gravity  of  mercury  is  13.6  times  that  of  pure  water, 
and  as  30  inches  is  2.5  feet,  13.6  X  2.5  =  34.0  feet.  A  water 
barometer  is  impracticable  for  use  in  measuring  atmospheric 
pressures,  but  it  is  convenient  to  know  its  approximate  height 
corresponding  to  a  given  height  of  the  mercury  barometer. 
The  following  table  gives  in  the  first  column  heights  of  the 
mercury  barometer,  in  the  second  the  corresponding  pressures 
per  square  inch,  in  the  third  the  pressures  in  atmospheres,  and 
in  the  fourth  the  heights  of  the  water  barometer.  This  fourth 
column  is  computed  by  multiplying  the  numbers  in  the  first 
column  by  1.133,  which  is  one-twelfth  of  13.6,  the  specific 
gravity  of  mercury. 

TABLE   II.     ATMOSPHERIC    PRESSURE. 


Mercury 
Barometer. 
Inches. 

"Pressure. 
Pounds  per 
Square  Inch. 

Pressure. 
Atmospheres. 

•   Water 
Barometer. 
Feet. 

Elevations. 
Feet. 

Boiling-point 
of  Water 

(Fahrenheit). 

31 

15-2 

1.03 

35-1 

-895 

213°.  9 

30 

14.7 

I. 

34-0 

o 

212    .2 

29 

14.2 

0.97 

32.9 

+  925 

2IO    .4 

28 

T3-7 

0-93 

31-7 

1880 

208    .7 

27 

13-2 

0.90 

30.6 

2870 

2O6    .9 

26 

12.7 

0.86 

29-5 

3900 

205   .0 

25 

12.2 

0.83 

28.3 

4970 

2O3     .1 

24 

ii.  7 

0.80 

27.2 

6085 

2OI     .  I 

23 

ii.  3 

0.76 

26.1 

7240 

199  .0 

22 

10.8 

0.72 

24.9 

8455 

196    .9 

21 

10.3 

0.69 

23-8 

9720 

194  .7 

20 

9.8 

0.67 

22.7 

11050 

192  .4 

This  table  also  gives  in  the  fifth  column  values  adapted 
from  the  vertical  scale  of  altitudes  used  in  barometric  work, 
which  show  approximate  vertical  heights  corresponding  to 


ART.  5.]  COMPRESSIBILITY  OF    WATER.  9 

barometer  readings,  provided  that  the  pressure  at  sea  level  is 
30  inches.*  In  the  last  column  are  given  the  approximate 
boiling-points  of  water  corresponding  to  the  readings  of  the 
mercury  barometer. 

Prob.  5.  What  pressure  in  pounds  per  square  inch  exists  at 
the  base  of  a  column  of  water  170  feet  high?  What  pressure 
in  atmospheres  ? 

ARTICLE  5.  COMPRESSIBILITY  OF  WATER. 

The  popular  opinion  that  water  is  incompressible  is  not 
justified  by  experiments,  which  show  in  fact  that  it  is  more 
compressible  than  iron  or  even  timber  within  the  elastic  limit. 
These  experiments  indicate  that  the  amount  of  compression  is 
directly  proportional  to  the  applied  pressure,  and  that  water  is 
perfectly  elastic,  recovering  its  original  form  on  the  removal  of 
the  pressure.  The  amount  of  linear  compression  caused  by  a 
pressure  of  one  atmosphere  is,  according  to  the  measures  of 
GRASSI,  from  0.000051  at  35°  Fahrenheit  to  0.000045  at  80° 
Fahrenheit. 

Taking  0.00005  as  a  rnean  value  of  the  linear  compression 
per  atmosphere,  the  coefficient  of  elasticity  of  water  is 

E  =  -  — —  =  294000  pounds  per  square  inch, 
0.00005 

which  is  only  one-fifth  of  the  coefficient  of  elasticity  of  timber, 
and  less  than  one-eightieth  that  of  wrought-iron.f 

A  column  of  water  hence  increases  in  density  from  the 
surface  downward.  If  its  weight  at  the  surface  be  62.5  pounds 
per  cubic  foot,  at  a  depth  of  34  feet  a  cubic  foot  will  weigh 

62.5  (i  -f  0.00005)  =  62.503  pounds, 

*  PLYMPTON,  The  Aneroid  Barometer  (New  York,  1878). 

f  MERRIMAN'S  Mechanics  of  Materials  (New  York,  1885),  p.  9. 


10  INTRODUCTION.  [CHAP.  I. 

and  at  a  depth  of  340  feet  a  cubic  foot  will  weigh 
62.5  (i  +  0.0005)  =  62.53  pounds. 

The  variation  in  weight,  due  to  compressibility,  is  hence  too 
small  to  be  regarded  in  hydrostatic  computations. 

Prob.  6.  If  w  be  the  weight  of  water  per  cubic  foot  at 
the  surface,  show  that  the  weight  at  a  depth  of  d  feet  is 
w  (i  +  0.0000015  d). 

ARTICLE  6.  THE  ACCELERATION  OF  GRAVITY. 

The  symbol  g  is  used  in  hydraulics  to  denote  the  accelera- 
tion of  gravity  ;  that  is,  the  increase  in  velocity  per  second  for 
a  body  falling  freely  in  a  vacuum  at  the  surface  of  the  earth. 
At  the  end  of  t  seconds  from  the  beginning  of  the  fall,  the 
velocity  of  the  body  is 


The  space,  h,  passed  over  in  this  time,  is  the  product  of  the 
mean  velocity,  \  V,  and  the  number  of  seconds,  /,  or 


The  relation  between  the  velocity  and  the  space  is  found  by 
eliminating  /  from  these  two  equations,  and  is 


Hence  the  velocity  of  a  body  which  has  fallen  freely  through 
any  height  varies  as  the  square  root  of  that  height.  This  equa- 
tion may  also  be  written  in  the  form 

,    v 

h  =  —  , 
If' 

which  shows  that  the  height  or  space  varies  with  the  square  of 
the  velocity  of  the  falling  body. 


ART.  6.]  THE  ACCELERATION  OF  GRAVITY.  II 

The  quantity  32.2  feet  per  second  per  second  is  an  approxi- 
mate value  of  g  which  is  often  used  in  hydraulic  formulas.  It 
is,  however,  well  known  that  the  force  of  gravity  is  not  of  con- 
stant intensity  over  the  earth's  surface,  but  is  greater  at  the  poles 
than  at  the  equator,  and  also  greater  at  the  sea  level  than  on 
high  mountains.  The  following  formula  of  PEIRCE,  *  which  is 
partly  theoretical  and  partly  empirical,  gives  the  value  of  g  in 
feet  for  any  latitude  /,  and  any  elevation  e  above  the  sea  level, 
e  being  taken  in  feet  : 

g  =  32.0894  (i  +  0.0052375  sin2  /)(i  —  0.000000095  7^)  : 
and  from  this  its  value  may  be  computed  for  any  locality. 

The  greatest  value  of  g  is  at  the  sea  level  at  the  pole,  for 
which 

/  =  90°,     e  =  o,  whence  g  ==  32.258. 


The  least  value  of^-is  on  high  mountains  at  the  equator;  for 
this  there  may  be  taken 

/=  o°,       e  =  10000  feet,     whence  g-=.  32.059. 
Again,  for  the  United  States  the  practical  limiting  values  are  : 

/=49°,     *  =  o,  whence  g—  32.186; 

/=  25°,     e  =  loooo  feet,     whence  g  =  32.089. 

These  results  indicate  that  32.2  feet  is  too  large  for  a  mean 
value  of  the  acceleration. 

In  the  numerical  work  of  this  book,  the  value  of  the  accel- 
eration is  taken  to  be,  unless  otherwise  stated, 

g  =  32.16  feet  per  second  per  second, 
*  SMITH'S  Hydraulics,  p.  19,  where  may  be  found  a  table  giving  values  of 


12  INTRODUCTION.  [CHAP.  I. 

from  which   the  frequently  occurring  quantity  V2g   is  found 
to  be 

=  8.02. 


If  greater  precision  be  required,  which  will  rarely  be  the  case, 
g  can  be  computed  from  the  formula  for  the  particular  latitude 
and  elevation  above  sea  level. 

Prob.  7.  Compute  the  value  of  g  for  the  latitude  40°  36', 
and  the  elevation  400  feet. 

Prob.  8.  What  is  the  value  of  g  if  the  unit  of  time  be  one 
minute?  Ans.  115  776  feet  per  minute  per  minute. 


ARTICLE  7.  NUMERICAL  COMPUTATIONS. 

The  numerical  work  of  computation  should  not  be  carried 
to  a  greater  degree  of  refinement  than  the  data  of  the  problem 
warrant.  For  instance,  in  questions  relating  to  pressures,  the 
data  are  uncertain  in  the  third  significant  figure,  and  hence 
more  figures  than  three  or  four  in  the  final  result  must  be 
delusive.  Thus,  let  it  be  required  to  compute  the  number  of 
pounds  of  water  in  a  box  containing  307.37  cubic  feet.  Taking 
the  mean  value  62.5  pounds  as  the  weight  of  one  cubic  foot, 
the  multiplication  gives  the  result  19210.625  pounds,  but 
evidently  the  decimals  here  have  no  precision,  since  the  last 
figure  in  62.5  is  not  accurate,  and  is  likely  to  be  less  than  5,  de- 
pending upon  the  impurity  of  the  water  and  its  temperature* 
The  proper  answer  to  this  problem  is  19  200  pounds,  or  per- 
haps  19210  pounds,  and  this  is  to  be  regarded  as  a  probable 
average  result  rather  than  an  exact  definite  quantity. 

The  use  of  logarithms  is  to  be  recommended  in  hydraulic 
computations,  as  thereby  both  mental  labor  and  time  are  saved. 
Four-figure  tables  are  sufficient  for  all  common  problems,  and 
their  use  is  particularly  advantageous  in  cases  where  the  data 
are  not  precise,  as  thus  the  number  of  significant  figures  in 


ART.  7-]  NUMERICAL   COMPUTATIONS.  13 

results  is  kept  at  about  three  and  statements  implying  great 
precision,  when  none  really  exists,  are  prevented.  In  some 
problems  five-figure  logarithms  will  be  needed,  but  probably  no 
hydraulic  data  are  ever  sufficiently  exact  to  require  the  use  of 
a  seven-figure  table.  Six-figure  logarithms  should  not  be  em- 
ployed if  others  can  be  obtained,  as  their  arrangement  is  not 
generally  convenient  for  interpolation. 

As  this  book  is  mainly  intended  for  the  use  of  students  in 
technical  schools,  a  word  of  advice  directed  especially  to  them 
may  not  be  inappropriate.  It  will  be  necessary  for  students  in 
order  to  gain  a  clear  understanding  of  hydraulic  science,  or  of 
any  other  engineering  subject,  to  solve  many  numerical  prob- 
lems, and  in  this  a  neat  and  systematic  method  should  be  cul- 
tivated. The  practice  of  performing  computations  on  any  loose 
scraps  of  paper  that  may  happen  to  be  at  hand  should  not  be 
followed,  but  the  work  should  be  done  in  a  special  book  pro- 
vided for  that  purpose,  and  be  accompanied  by  such  explanatory 
remarks  as  may  seem  necessary  in  order  to  render  the  solution 
clear.  Such  a  note-book,  written  in  ink,  and  containing  the 
fully  worked  out  solutions  of  the  problems  and  examples  given 
in  these  pages,  will  prove  of  great  value  to  every  student  who 
makes  it. 

Prob.  9.  Compute  the  weight  of  a  column  of  water  1.1286 
inches  in  diameter  and  34.0  feet  high  at  the  temperature  of 
62°  Fahrenheit. 

Prob.  10.  How  many  gallons  of  water  are  contained  in  a 
pipe  4  inches  in  diameter  and  12  feet  long?  How  many 
pounds? 


14  HYDROSTATICS.  [CHAP.  II% 


CHAPTER  II. 
HYDROSTATICS. 

ARTICLE  8.  TRANSMISSION  OF  PRESSURES. 

One  of  the  most  remarkable  properties  of  water  is  its 
capacity  of  transmitting  a  pressure,  applied  at  one  point  of  the 
surface  of  a  closed  vessel,  unchanged  in  intensity,  in  all  direc- 
tions, so  that  the  effect  of  the  applied  pressure  is  to  cause  an 
equal  force  per  square  inch  upon  all  parts  of  the  enclosing  sur- 
face. This  is  a  consequence  of  the  perfect  fluidity  of  the  water, 
by  which  its  particles  move  freely  over  each  other  and  thus 
transmit  the  applied  pressure. 

An  experimental  proof  of  this  property  is  seen  in  the  hydro- 
static press,  where  the  force  applied  to  the  small  piston  is  ex- 
erted through  the  fluid  and  produces  an  equal  unit-pressure 
at  every  point  on  the  large  piston.  The  applied  force  is  here 
multiplied  to  any  required  extent,  but  the  work  performed  by 
the  large  piston  cannot  exceed  that  imparted  to  the  fluid  by 
the  small  one.  Let  ^i  and  A  be  the  areas  of  the  small  and 
large  pistons,  and  /  the  pressure  in  pounds  per  square  unit  ap- 
plied to  a ;  then  the  total  pressure  on  the  small  piston  is  pa> 
and  that  on  the  large  piston  is pA.  Let  the  distances  through 
which  the  pistons  move  at  one  stroke  of  the  smaller  be  d  and 
D.  Then  the  imparted  work  \spad,  and  the  performed  work, 
neglecting  hurtful  resistances,  \spAD.  Consequently  ad  =  AD> 
and  since  a  is  small  as  compared  with  A,  the  distance  D  must 


ART.  9-]  HEAD  AND  PRESSURE.  1 5 

be  small  compared  with  d.     Here  is  found  an  illustration  of  the 
popular  maxim  that  "  What  is  gained  in  force  is  lost  in  velocity." 

The  pressure  existing  at  any  point  within  a  body  of  water 
is  exerted  in  all  directions  with  equal  intensity.  This  im- 
portant property  follows  at  once  from  that  of  the  transmission 
of  pressure,  for  this  may  be  regarded  as  effected  by  the  con- 
fined body  of  water  acting  as  an  elastic  spring  which  presses 
outwards  in  all  directions.  Thus  every  particle  of  the  water  is 
in  a  state  of  stress,  and  reacts  in  all  directions  with  equal  in- 
tensity. And  the  same  principle  applies  to  a  particle  within  a 
body  of  water  whose  surface  is  free,  for  the  pressure  which  ex- 
ists at  any  point  due  to  the  weight  above  it  produces  a  state  of 
stress  among  all  the  fluid  particles. 

Prob.  II.  In  a  hydrostatic  press  a  work  of  one-fourth  a 
horse-power  is  applied  to  the  small  piston.  The  diameter  of 
the  large  piston  is  12  inches,  and  it  moves  half  an  inch  per 
minute.  Find  the  pressure  per  square  inch  in  the  fluid. 

Ans.  1750  pounds. 

ARTICLE  9.  HEAD  AND  PRESSURE. 

The  free  surface  of  water  at  rest  is  perpendicular  to  the  di- 
rection of  the  force  of  gravity,  and  for  bodies  of  water  of  small 
extent  this  surface  may  be  regarded  as  a  plane.  Any  depth 
below  this  plane  is  called  "a  head,"  or  the  head  upon  any 
point  is  its  vertical  depth  below  the  level  surface.  Let  h  be 
the  head  and  w  the  weight  of  a  cubic  unit  of  water;  then  at 
the  depth  h  one  horizontal  square  unit  bears  a  pressure  equal 
to  the  weight  of  a  column  of  water  whose  height  is  //,  and 
whose  cross-section  is  one  square  unit,  or  wh.  But  the  pres- 
sure at  this  point  is  exerted  in  all  directions  with  equal  inten- 
sity. The  unit-pressure  /  at  the  depth  h  then  is 

p  =  wh\ (I) 


16  HYDROSTATICS.'  [CHAP.  II. 

and  conversely  the  depth,  or  head,  for  a  unit-pressure  /  is 

k  =  P.  (i') 

w  \    * 

If  h  be  taken  in  feet  and  /  in  pounds  per  square  foot,  these 
formulas  are 

/  =  62.5^, 
h  =  o.oi6/. 

Hence  pressure  and  head  are  mutually  convertible,  and  in  fact 
one  is  often  used  as  synonymous  with  the  other,  although  really 
each  is  proportional  to  the  other.  Any  pressure  /  can  be  re- 
garded as  produced  by  a  head  h,  which  sometimes  is  called  the 
"  pressure  head." 

In  numerical  work  p  is  usually  taken  in  pounds  per  square 
inch,  while  h  is  expressed  in  feet.  Thus,  the  pressure  in  pounds 
per  square  foot  is  62.5/2,  and  the  pressure  in  pounds  per  square 
inch  is  yj^  of  this  ;  or, 

/  =  0.434^, 
h  =  2.304^. 

Stated  in  words  these  rules  are : 

I  foot  head  corresponds  to  0.434  pounds  per  square  inch ; 
I  pound  per  square  inch  corresponds  to  2.304  feet  head. 

These  values,  be  it  remembered,  depend  upon  the  assump- 
tion that  62.5  pounds  is  the  weight  of  a  cubic  foot  of  water, 
and  hence  are  liable  to  variation  in  the  third  significant  figure 
(Art.  3).  The  extent  of  these  variations  for  fresh  water  may 
be  judged  by  the  following  table,  which  gives  multiples  of  the 
above  values,  and  also  the  corresponding  quantities  when  the 
cubic  foot  is  taken  as  62.3  pounds. 


ART.  9.J  HEAD  AND  PRESSURE. 

TABLE  III.   HEADS  AND  PRESSURES. 


Head 

Pressure  in  Pounds 
per  Square  Inch. 

Pressure 
in  Pounds 

Head  in  Feet. 

in  Feet. 

per  Square 
Inch. 

•w  =  62.5 

•w  =  62.3 

w  =  62.5 

iv  =  62.3 

I 

0-434 

0.433 

I 

2.304 

2.3II 

2 

0.868 

0.865 

2 

4.608 

4.623 

3 

1.302 

1.298 

3 

6.912 

6-934 

4 

1.736 

I-73I 

4 

9.216 

9.246 

5 

2.  I7O 

2.  163 

5 

11.520 

11-557 

6 

2.604 

2.596 

6 

I3.824 

13.868 

7 

3-038 

3.028 

7 

16.128 

16.180 

8 

3-472 

3.461 

8 

18.432 

18.491 

9 

3-906 

3.894 

9 

20.736 

20.803 

10 

4-340 

4.326 

10 

23.040 

23.114 

The  atmospheric  pressure,  whose  average  value  is  14.7  pounds 
per  square  inch,  is  transmitted  through  water,  and  is  to  be  added 
to  the  pressure  due  to  the  head  whenever  it  is  necessary  to 
regard  the  absolute  pressure.  This  is  important  in  some  in- 
vestigations on  the  pumping  of  water,  and  in  a  few  other  cases 
where  a  partial  or  complete  vacuum  is  produced  on  one  side  of 
a  body  of  water.  For  example,  if  the  air  be  exhausted  from  a 
small  globe,  so  that  its  tension  is  only  5  pounds  per  square 
inch,  and  it  be  submerged  in  water  to  a  depth  of  250  feet,  the 
absolute  pressure  per  square  inch  on  the  globe  is 

/  =  0.434  X  250+  14.7  =  123.2  pounds, 
and  the  resultant  effective  pressure  per  square  inch  is 
•p'  ~  123.2—  5.0  =  118.2  pounds. 

Unless  otherwise  stated,  however,  the  atmospheric  pressure 
need  not  be  regarded,  since  under  ordinary  conditions  it  acts 
with  equal  intensity  upon  both  sides  of  a  submerged  surface. 

Prob.  12.  What  unit  pressure  corresponds  to  230  feet  head  ? 
What  head  in  meters  produces  a  pressure  of  10  kilograms  per 
square  centimeter? 


1 8  HYDROSTATICS.  [CHAP.  II 

Prob.  13.  A  bottle  contains  air  of  2  pounds  per  square  inch 
tension,  and  its  cork,  0.75  inches  in  diameter,  can  be  forced  in 
by  a  pressure  of  100  pounds.  How  deep  under  water  must  it 
be  sunk  in  order  to  force  the  cork?  Ans.  492  feet. 

ARTICLE  10.  NORMAL  PRESSURE. 

The  total  normal  pressure  on  any  submerged  surface  may  be 
found  by  the  following  theorem  : 

The  normal  pressure  is  equal  to  the  product  of  the  weight 
of  a  cubic  unit  of  water,  the  area  of  the  surface,  and  the 
head  on  its  centre  of  gravity. 

To  prove  this  let  A  be  the  area  of  the  surface,  and  imagine 
'  it  to  be  composed  of  elemen- 

tary areas,  alf  az,  a3,  etc., 
each  of  which  is  so  small  that 
the  unit-pressure  over  it  may 
be  taken  as  uniform ;  let 
/it ,  h^ ,  7^3 ,  etc.,  be  the  heads 
FlG- 2-  on  these  elementary  areas, 

and  let  w  denote  the  weight  of  a  cubic  unit  of  water.  The 
unit-pressures  at  the  depths  /^  ,  h^ ,  //, ,  etc.,  are  w/it ,  wh^ ,  wh^ , 
etc.  (Art.  9),  and  hence  the  normal  pressures  on  the  elementary 
areas  al9  az,a3,  etc.,  are  waji^ ,  wajt^ ,  waji^ ,  etc*  The  total 
normal  pressure  P  on  the  entire  surface  then  is 

P  =  w(alhl  +  tf  A  +  aji3  +  etc.). 

Now  let  h  be  the  head  on  the  centre  of  gravity  of  the  surface  • 
then,  from  the  definition  of  the  centre  of  gravity, 

#  A  +  ^  A  +  aA  +  etc-  —  Ah* 
Therefore  the  normal  pressure  is 

P  =  wAh, (2) 

which  proves  the  theorem  as  stated. 


ART.  zo.J  NORMAL  PRESSURE.  19 

This  rule  applies  to  all  surfaces,  whether  plane,  curved,  or 
warped,  and  however  they  be  situated  with  reference  to  the 
water  surface.  Thus  the  total  normal  pressure  upon  the  sur- 
face of  a  submerged  cylinder  remains  the  same  whatever  be  its 
position,  provided  the  depth  of  the  centre  of  gravity  of  that 
surface  be  kept  constant.  It  is  best  to  take  h  in  feet,  A  in 
square  feet,  and  w  as  62.5  ;  then  Pwill  be  in  pounds.  In  case 
surfaces  are  given  whose  centres  of  gravity  are  difficult  to  de- 
termine, they  should  be  divided  into  simpler  surfaces,  and  then 
the  total  normal  pressure  is  tke  sum  of  the  normal  pressures  on 
the  separate  surfaces. 

The  normal  pressure  on  the  base  of  a  vessel  filled  with  water 
is  equal  to  the  weight  of  a  cylinder  of  water  whose  base  is  the 
base  of  the  vessel,  and  whose  height  is  the  depth  of  water,  and 
only  in  the  case  of  a  vertical  cylinder  does  this  become  equal  to 
the  weight  of  the  water.  Thus  the  pressure  on  the  base  of  a 
vessel  depends  upon  the  depth  of  water  and  not  upon  the 
shape  of  the  vessel.  Also  in  the  case  of  a  dam,  the  depth  of 
the  water  and  not  the  size  of  the  pond  determines  the  amount 
of  pressure. 

The  normal  pressure  on  the  interior  surface  of  a  sphere  filled 
with  water  is  greater  than  the  weight  of  the  water,  for  the 
weight  acts  only  vertically,  while  the  normal  pressures  are  ex- 
erted in  all  directions.  If  d  be  the  diameter  of  the  sphere,  for- 
mula (2)  gives 


while  the  weight  of  water  is 
W= 


Hence  the  interior  normal  pressure  is  three  times  the  weight 
of  the  water. 

Prob.  14.  A  cone  with  altitude  h  and  diameter  of  base  d  is 
filled  with  water.     Find  the  normal  pressure  on  the  interior 


2O  HYDROSTATICS.  [CHAP.  II. 

surface  (a)  when  it  is  held  vertical  with  base  downward  ;  (b)  when 
held  horizontal. 

Prob.  15.  A  board  2  feet  wide  at  one  end  and  2  feet  6 
inches  at  the  other  is  8  feet  long.  What  is  the  normal  pres- 
sure upon  each  of  its  sides  when  placed  vertically  in  water  with 
the  narrow  end  in  the  surface? 

ARTICLE  n.  PRESSURE  IN  A  GIVEN  DIRECTION. 

The  pressure  against  a  submerged  plane  surface  in  a  given 
direction  may  be  iound  by  obtaining  the  normal  pressure  by 
Art.  10  and  computing  its  component  in  the  required  direc- 
tion, or  by  means  of  the  following  theorem  : 

The  horizontal  pressure  on  any  plane  surface  is  equal  to 
the  normal  pressure  on  its  vertical  projection ;  the 
vertical  pressure  is  equal  to  the  normal  pressure  on  its 
horizontal  projection  ;  and  the  pressure  in  any  direction 
is  equal  to  the  normal  pressure  on  a  projection  perpen- 
dicular to  that  direction. 

To  prove  this  let  A  be  the  area  of  the  given  surface,  repre- 
sented by  AA  in  Fig.  3,  and 
P  the  normal  pressure  upon 
it,  or  P  =  wAh.  Now  let  it 
be  required  to  find  the  pres- 
sure P'  in  a  direction  mak- 
ing an  angle  6  with  the 
normal  to  the  given  plane. 
Draw  A' A'  perpendicular  to 
.the  direction  of  Prt  and  let 

A'  be  the  area  of  the  projection  of  A  upon  it.     The  value  of 

P'  then  is 

P'  =  P  cos  d  =  wAh  cos  6. 

But  A  cos  6  is  the  value  of  A'  by  the  construction.     Hence 
P.-wA't, (3) 

and  the  theorem  is  thus  demonstrated. 


ART.  ii. J  PRESSURE  IN  A    GIVEN  DIRECTION.  21 

This  theorem  does  not  in  general  apply  to  curved  surfaces. 
But  in  cases  where  the  head  of 
water  is  so  great  that  the  pressure 
may  be  regarded  as  uniform  it  is 
also  true  for  curved  surfaces.  For 
instance,  consider  a  cylinder  or 
sphere  subjected  on  every  ele- 
mentary area  to  the  unit-pressure 
p  due  to  the  high  head  h,  and  let 
it  be  required  to  find  the  pressure 
in  the  direction  shown  by  ql ,  q^ , 

and  ^,  in  Fig.  4.  The  pressures  A » A »  /3 ,  etc.,  on  the  ele- 
mentary areas  al ,  #2 ,  aa ,  etc.,  are 

A  =  Pai ,    A  =  Pa*  >    A  =  M ,     etc., 
and  the  components  of  these  in   the  given  direction  are 

^  =  pal  cos  8, ,     q^  —  pa^  cos  03 ,     qz  =  paz  cos  03 ,   etc., 
whence  the  total  pressure  P'  in  the  given  direction  is 
P'  —  p(a^  cos  6l  +  #2  cos  6>2  +  a,  cos  <93  +  etc.). 

But  the  quantity  in  the  parenthesis  is  the  projection  of  the  sur- 
face on  a  plane  perpendicular  to  the  given  direction,  or  MN. 
Hence 

P  —  p  x  area  MN, 

which  is  the  same  rule  as  for  plane  surfaces. 

For  the  case  of  a  water-pipe  let  /  be  the  interior  pressure  per 
square  inch,  and  d  its  diameter  in  inches.  Then  for  a  length 
of  one  inch  the  force  tending  to  rupture  the  pipe  longitudinally 
is  pd.  This  is  resisted  by  the  unit  stress  5  in  the  walls  of  the 
pipe  acting  over  the  area  2t,  if  t  be  the  thickness.  As  these 
forces  are  equal, 

2St  =  pd, 


22 


HYDROSTATICS. 


[CHAP.  II. 


which  is  the  fundamental  equation  for  the  discussion  of  the 
strength  of  water-pipes. 

Prob.  1 6.  The  back  of  a  dam  has  a  slope  of  \\  to  I.  Find 
the  horizontal  pressure  per  linear  foot  upon  it,  the  water  being 
13  feet  deep. 

Prob.  17.  What  head  of  water  will  burst  a  pipe  24  inches 
in  interior  diameter  and  0.75  inches  thick,  the  tensile  strength 
of  the  cast-iron  being  20,000  pounds  per  square  inch  ? 


ARTICLE  12.  CENTRE  OF  PRESSURE  ON  RECTANGLES. 

The  centre  of  pressure  on  a  surface  submerged  in  water  is 
the  point  of  application  of  the  resultant  of  all  the  normal  pres- 
sures upon  it.  The  simplest  and  probably  the  most  important 
case  is  the  following : 

If  a  rectangle  be  placed  with  one  end  in  the  water  surface, 
the  centre  of  pressure  is  distant  from  that  end  two- 
thirds  of  its  length. 

This  theorem  will  be  proved  by  the  help  of  the  graphical 
illustration  shown  in  Fig.  5.  The  rectangle,  which  in  practice 

might  be  a  board,  is  placed  with 
its  breadth  perpendicular  to  the 
plane  of  the  drawing,  so  that 
AB  represents  its  edge.  It  is 
required  to  find  the  centre  of 
pressure  C.  For  any  head  h  the 
unit-pressure  is  wh  (Art.  9),  and 
hence  the  unit-pressures  on  one 
side  of  AB  may  be  graphically 
represented  by  arrows  which  form  a  triangle.  Now  if  a  force 
P  equal  to  the  total  pressure  is  applied  on  the  other  side  of 
the  rectangle  to  balance  these  unit-pressures,  it  must  be  placed 
opposite  to  the  centre  of  gravity  of  the  triangle.  Therefore 


FIG.  5. 


ART.  12.]      CENTRE   OF  PRESSURE   ON  RECTANGLES.  2$ 

A  C  equals  two-thirds  of  AB,  and  the  rule  is  proved.     The  head 
on  C  is  evidently  also  two-thirds  of  the  head  on  B. 

Another  case  is  that  shown  in  Fig.  6,  where  the  rectangle, 
whose  length  is  B^Bt ,  is  wholly  immersed,  the  head  on  B^  being 
//, ,  and  on  B^  being  hn_.  Let  A 


=  b^  Now  the  normal  pres- 
sure P1  on  ABl  is  applied  at 
the  distance  f£,  from  At 
and  the  normal  pressure  P^ 

on  AB^  is  applied  at  the  dis-  FlG>  6* 

tance  f£2  from  A.     The  normal  pressure  P  on  i 
sultant  of  Pl  and  Pt ,  or 

P  =  P,-P1', 
and  also,  by  taking  moments  about  A  as  a  centre, 


Now,  by  Art.  10,  the  values  of  Pt  and  Pl  are,  for  a  rectangle  one 
unit  in  breadth, 


hence 


z  =  w  X  b,  X  K, 
P  = 


t  X  \h,  ; 


and  inserting  these  in  the  equation  of  moments,  the  value  of 
y  is 


Now  if  6  be  the  angle  of  inclination  of  the  plane  to  the  water 
surface,  7/2  =  £2  sin  0,  and  hl  =  b^  sin  0.  Accordingly,  the  ex- 
pression becomes 


'*.'-  *.' 


24  HYDROSTATICS.  [CHAP.  II. 

Again,  if  h'  be  the  head  on  the  centre  of  pressure,  y  =  h'  cosec  0, 
b^  •=.  //a  cosec  6,  and  bl  =  /^  cosec  6.  These  inserted  in  the 
last  equation  give 

j.j  - 2  *.'  -  v 

3  A* -4" 

These  formulas  are  very  convenient  for  computation,  as  the 
squares  and  cubes  may  be  taken  from  tables. 

If  hl  equals  h^  the  above  formula  becomes  indeterminate, 
which  is  due  to  the  existence  of  the  common  factor  //a  —  /^  in 
both  numerator  and  denominator  of  the  fraction  ;  dividing  out 
this  common  factor,  it  becomes 

t,_2*.'  +  ^.  +  *.' 

"3         A.  +  At 
from  which,  if  /*3  ^hl  —  h,  there  is  found  the  result  h!  =  h. 

If /^r=o,  or  bl  =  o,^y  becomes  f#a  and  h'  becomes  -£/ja, 
which  proves  again  the  special  rule  given  at  the  beginning  of 
this  article. 

Prob.  1 8.  A  rectangle  4  feet  long  is  immersed  in  water  with 
its  ends  parallel  to  the  surface,  the  head  on  one  end  being  7  feet 
and  that  on  the  other  9  feet.  Find  the  head  on  the  centre  of 
pressure,  and  also  the  value  of  P. 

ARTICLE  13.  GENERAL  RULE  FOR  CENTRE  OF  PRESSURE. 

For  any  plane  surface  submerged  in  a  liquid,  the  centre  of 
pressure  may  be  found  by  the  following  rule  : 

Find  the  moment  of  inertia  of  the  surface  and  its  statical 
moment,  both  with  reference  to  an  axis  situated  at  the 
intersection  of  the  plane  of  the  surface  with  the  water 
level.  Divide  the  former  by  the  latter,  and  the  quotient 
is  the  perpendicular  distance  from  that  axis  to  the 
centre  of  pressure. 


ART.  13.]    GENERAL   RULE   FOR   CENTRE   OF  PRESSURE.  2$ 

The  demonstration  is  analogous  to  that  in  the  last  article. 
Let,  in  Fig.  6,  BJB^  be  the  trace  of  the  plane  surface,  which 
itself  is  perpendicular  to  the  plane  of  the  drawing,  and  C  be 
the  centre  of  pressure,  at  a  distance  y  from  A  where  the  plane 
of  the  surface  intersects  the  water  level.  Let  al ,  az ,  a3 ,  etc.,  be 
elementary  areas  of  the  surface,  and  /*, ,  //„ ,  //8 ,  etc.,  the  heads 
upon  them,  which  produce  the  normal  elementary  pressures, 
wafa  ,  wajt^ ,  K'tfs//3 ,  etc.  Let  y^ ,  jj/2 ,  J3 ,  etc.,  be  the  distances 
from  A  to  these  elementary  areas.  Then  taking  the  point  A 
as  a  centre  of  moments,  the  definition  of  centre  of  pressure 
gives  the  equation 

(wajt^  +  wajii  +  wajtt  -\-  etc.)  y  = 

wa^h^y^  -f  waji^y^  +  wa^ys  +  etc. 

Now  let  6  be  the  angle  of  inclination  of  the  surface  to  the 
water  level ;  then  h,  =  y^  sin  0,  h^  =/2sin  0,  hz  =  j/8sin  0,  etc. 
Hence,  inserting  these  values,  the  expression  for^  is 

etc. 


etc. 

The  numerator  of  this  fraction  is  the  sum  of  the  products 
obtained  by  multiplying  each  element  of  the  surface  by  the 
square  of  its  distance  from  the  axis,  which  is  called  the  moment 
of  inertia  of  the  surface.  And  the  denominator  is  the  sum  of 
the  products  of  each  element  of  the  surface  by  its  distance  from 
the  axis,  which  is  called  the  statical  moment  of  the  surface. 
Therefore 

_  moment  of  inertia  _  /' 
•^  '       statical  moment     ~~  5 

is  the  general  rule  for  finding  centres  of  pressure  for  plane 
surfaces. 

The  statical  moment  of  a  surface  is  simply  its  area  multi- 
plied by  the  distance  of  its  centre  of  gravity  from  the  given 


26  HYDROSTATICS.  [CHAP.  II. 

axis,  as  is  evident  from  the  definition  of  centre  of  gravity.  The 
moments  of  inertia  of  plane  surfaces  with  reference  to  an  axis 
through  the  centre  of  gravity  are  deduced  in  works  on  theo- 
retical mechanics  ;  a  few  values  are  : 

bd* 
For  a  rectangle  of  breadth  b  and  depth  d,   I  =  -  ; 

7      73 

For  a  triangle  with  base  b  and  altitude  d,    I  =  —  >-  ; 

For  a  circle  with  diameter  d,  I  =  -?—  . 

64 

To  find  from  these  the  moment  of  inertia  with  reference  to  a 
parallel  axis,  the  well-known  formula  •/'  =  I-\-Ak2  is  to  be  used, 
where  A  is  the  area  of  the  surface  and  k  the  distance  from  the 
given  axis  to  the  centre  of  gravity  of  the  surface,  and  /'  the 
moment  of  inertia  required. 

For  example,  let  it  be  required  to  find  the  centre  of  pres- 
sure of  a  circle  which  is  submerged  with  one  edge  in  the  water 
surface.  The  area  of  the  circle  is  \7td*,  and  its  statical  moment 
with  reference  to  the  upper  edge  is  \nd*  X  %d.  Then  from  (4), 

Ttd*       Ttd^    d* 

6J-  +  T"'4"   =5_^. 
y'  *  ' 


hence  the  centre  of  pressure  of  a  circle  with  one  edge  in  the 
water  surface  is  at  \d  below  the  centre.  Again,  for  a  triangle 
submerged  with  its  vertex  in  the  water  surface, 

b 


bd    2d 

~2~'T 


ART.  14.]    PRESSURES  ON  OPPOSITE  SIDES  OF  A  PLANE.          2/ 

Prob.  19.  Find  the  centre  of  pressure  of  the  triangle  in 
Fig.  9  when  it  is  inverted  so  that  its  base  is  in  the  surface. 

Prob.  20.  Find  the  centre  of  pressure  of  a  circle  when  verti- 
cally submerged  in  water  so  that  the  head  on  its  centre  is  equal 
to  two  diameters  of  the  circle.  Ans.  2.03^. 

ARTICLE  14.  PRESSURES  ON  OPPOSITE  SIDES  OF  A  PLANE. 

In  the  case  of  an  immersed  plane  the  water  presses  equally 
upon  both  sides  so  that  no  disturbance  of  the  equilibrium  re- 
sults from  the  pressure.  But  in  case 
the  water  is  at  different  levels  on  op- 
posite sides  of  the  surface  the  opposing 
pressures  are  unequal.  For  example, 
the  cross-section  of  a  self-acting  tide- 
gate,  built  to  drain  a  salt  marsh,  is 
shown  in  Fig.  7.  On  the  ocean  side 
there  is  a  head  of  /^  above  the  sill, 
which  gives  for  every  linear  foot  of 
the  gate  the  pressure 


which  is  applied  at  the  distance  \h^  above  the  sill.     On  the 
other  side  the  head  on  the  sill  is  h^  ,  which  gives  the  pressure 


whose  centre  of  pressure  is  at  \h^  above  the  sill.     The  result- 
ant pressure  P  is 


and  if  z  be  the  distance  of  the  point  of  application  of  P  above 
the  sill,  the  equation  of  moments  is 

(P,  -  PJ*  =  Pl  X  ta  -  />,  X  te, 
from  which  z  can  be  computed. 


28 


HYDROSTATICS. 


[CHAP.  II- 


The  action  of  the  gate  in  resisting  the  water  pressure  is  like 
that  of  a  beam  under  its  load,  the  two  points  of  support  being 
at  the  sill  and  the  hinge.  If  h  be  height  of  the  gate,  the  reac- 
tion at  the  hinge  is, 


and  this  has  its  greatest  value  when  /^  becomes  equal  to  h, 
and  h^  is  zero.  In  the  case  of  the  vertical  gate  of  a  canal  lock, 
which  swings  horizontally  like  a  door,  a  similar  problem  arises 
and  a  similar  conclusion  results. 

Prob.  21.  If  the  head  on  one  side  of  a  tide-gate  is  7  feet 
and  on  the  other  4  feet,  find  the  resultant  pressure  and  its 
point  of  application  above  the  sill. 

Ans.  1031  pounds  per  linear  foot,  at  2.82  feet  above  sill. 

ARTICLE  15.  MASONRY  DAMS. 

The  preceding  articles  show  that  the  pressure  on  the  back 
of  a  masonry  dam  is  normal  to  that  surface  at  every  point.  If 

the  back  be  a  plane  surface  the 
resultant  pressure  is  normal  to 
the  plane,  and  its  point  of  applica- 
tion is  at  two-thirds  of  the  length 
from  the  water  level.  Thus  in 
Fig.  8,  A  C  is  two-thirds  of  AB.  If 
h  be  the  head  of  water  above  the 
base  of  the  dam,  and  V  be  the 
angle  of  inclination  of  the  plane  of 
the  back  to  the  vertical,  the  normal  pressure  per  linear  foot  of 
the  dam  is,  from  Art.  10, 

P  =  w  X  h  sec  0  x  %k  =  %wh*  sec  0, 

which  shows  that  the  total  pressure  against  the  dam  varies  as 
the  square  of  its  height.  The  horizontal  component  of  this 


FIG.  8. 


ART.  15.]  MASONRY  DAMS.  29 

pressure   is  -kc'//a,  which  is  the  same  as  the  normal  pressure 
against  a  wall  whose  back  is  vertical. 

It  is  not  the  place  here  to  enter  into  the  discussion  of  the 
subject  of  the  design  of  masonry  dams,  but  the  two  ways  in 
which  they  are  liable  to  fail  may  be  noted.  The  first  is  that  of 
sliding  along  a  horizontal  joint,  as  BD  ;  here  the  horizontal 
component  of  the  thrust  overcomes  the  resisting  force  of  friction 
acting  along  the  joint.  If  W  be  the  weight  of  masonry  above 
the  joint,  and  /the  coefficient  of  friction,  the  resisting  friction 
is  fW,  and  the  dam  will  slide  if  the  horizontal  component  of 
the  pressure  is  equal  to  or  greater  than  this.  The  condition 
for  failure  by  sliding  then  is 


The  second  method  of  failure  is  that  of  rotating  around  the 
toe  D\  this  occurs  when  the  moment  of  P  equals  the  moment 
of  PFwith  reference  to  that  point;  or  if  /  and  m  be  the  lever- 
arms  dropped  from  D  upon  the  directions  of  P  and  W,  the 
condition  for  failure  by  rotation  is 

Pl=  Wm. 

In  practice  the  joints  are  so  built  as  to  give  full  security 
against  sliding,  so  that  the  usual  method  of  failure  is  by 
rotation. 

As  an  example  of  the  application  of  these  principles  con- 
sider a  rectangular  vertical  masonry  dam  which  weighs  140 
pounds  per  cubic  foot,  and  which  is  4  feet  wide.  First,  let  it 
be  required  to  find  the  height  for  which  it  would  fail  by  slid- 
ing, the  coefficient  of  friction  being  0.75.  The  horizontal 
water  pressure  is  \  X  62.5  X  1?,  and  the  resisting  friction  is 
0.75  X  140  X  4  X  h.  Placing  these  equal,  there  is  found  h  — 
13.4  feet.  Secondly,  to  find  the  height  for  which  failure  will 
occur  by  rotation,  the  equation  of  moments  is  stated  with  ref- 


3O  HYDROSTATICS.  [CHAP.  II. 

erence  to  the  front  lower  edge,  the  lever-arm  of  the  pressure 
being  \h,  and  that  of  the  wall  2  feet.     Hence 

J  X  62.5  X  ??  X  \h  =  140  X  4  X  h  X  2, 
from  which  there  is  found  h  =  10.4  feet. 

Prob.  22.  A  dam  whose  cross-section  is  a  triangle  has  a 
vertical  back,  is  3  feet  wide  at  the  base,  and  15  feet  high. 
Find  the  height  to  which  the  water  may  rise  behind  it  in  order 
to  cause  failure  (a)  by  sliding,  and  ($)  by  rotation,  using  0.75 
for  the  coefficient  of  friction  and  140  pounds  per  cubic  foot  for 
the  weight  of  the  masonry. 

ARTICLE  16.  Loss  OF  WEIGHT  IN  WATER. 

It  is  a  familiar  fact  that  bodies  submerged  in  water  lose 
part  of  their  weight  .  a  man  can  carry  under  water  a  large 
stone  which  would  be  difficult  to  lift  in  air ;  timber  when  sub- 
merged has  a  negative  weight  or  tends  to  rise  to  the  surface. 
The  following  is  the  law  of  loss :  * 

The  weight  of  a  body  submerged  in  water  is  less  than  its 
weight  in  air  by  the  weight  of  a  volume  of  water  equal 
to  that  of  the  body. 

To  demonstrate  this,  consider  that  the  submerged  body 
is  acted  upon  by  the  water  pressure  in  all  directions,  and 
that  the  horizontal  components  of  these  pressures  must  bal- 
ance. Any  vertical  elementary  prism  is  subjected  to  an  up- 
ward pressure  upon  its  base  which  is  greater  than  the  down- 
ward pressure  upon  its  top,  since  these  pressures  are  due  to 
the  heads.  Let  h^  be  the  head  on  the  top  of  the  elementary 
prism  and  /^  that  on  its  base,  and  a  the  cross-section  of  the 
prism ;  then  the  downward  pressure  is  wah^  and  the  upward 
pressure  is  wah^.  The  difference  of  these,  wa(Ji^  —  //,)  is  the  re- 
sultant upward  water  pressure,  and  this  is  equal  to  the  weight 
of  a  column  of  water  whose  cross-section  is  a  and  whose  height 


*  Discovered  by  ARCHIMEDES,  about  250  B.C. 


ART.  17.]  'DEPTH  OF  FLOTATION.  31 

is  that  of  the  elementary  prism.  Extending  this  to  all  the 
elementary  prisms  which  make  up  the  body,  it  is  seen  that  the 
upward  water  pressure  diminishes  its  weight  by  the  weight  of 
a  volume  of  water  equal  to  that  of  the  body. 

It  is  important  to  regard  this  loss  of  weight  in  constructions 
under  water.  If,  for  example,  a  dam  of  loose  stones  allows  the 
water  to  percolate  through  it,  its  weight  per  cubic  foot  is  less 
than  its  weight  in  air,  so  that  it  can  be  more  easily  moved  by 
horizontal  forces.  As  stone  weighs  about  150  pounds  per  cubic 
foot  in  air,  its  weight  in  water  is  only  about  150—62  —  88 
pounds. 

Prob.  23.  A  bar  of  iron  one  square  inch  in  cross-section 
and  one  yard  long  weighs  10  pounds  in  air.  What  is  its  weight 
in  water  ? 

ARTICLE  17.  DEPTH  OF  FLOTATION. 

When  a  body  floats  upon  water  it  is  sustained  by  an  upward 
pressure  of  the  water  equal  to  its  own  weight,  and  this  pressure 
is  the  same  as  the  weight  of  the  volume  of  water  displaced  by 
the  body.  Let  W  be  the  weight  of  the  floating  body  in  air, 
and  Wbe  the  weight  of  the  displaced  water;  then 

W=IV. (5) 

Now  let  z  be  the  depth  of  flotation  of  the  body ;  then  to  find 
its  value  for  any  particular  case  W  is  to  be  expressed  in  terms 
of  the  linear  dimensions  of  the  body,  and  W'm  terms  of  the 
depth  of  flotation  z. 

For  example,  a  cone  which  weighs  w'  pounds  per  cubic  foot 

floats  with  its  base  downward  as 
represented  in  Fig.  9,  its  altitude 
being  d  and  the  radius  of  its  base  b. 
The  weight  of  the  floating  cone  is 

J        w  =  wf  •  nv  -  K 

FIG.  9.  and   the  weight   of  the   displaced 

water  is  that  of  a  frustum  of  the  altitude  z,  or 


32  HYDROSTATICS.  [CHAP.  II. 


3  3 

Equating  these  values  and  solving  for  z  gives  the  result 


which  is  the  depth  of  flotation.     If  w'  =  w,  the  cone  has  the 
same  density  as  water,  and  z  —  d\  if  w'  =  o,  the  cone  has  no 

weight,  and  z  =  o. 

To  find  the  depth  of  flotation 
for  a  cylinder  lying  horizontally, 
let  w'  be  its  weight  per  cubic 
foot,  and  r  the  radius  of  its  cross- 
section.  The  depth  of  flotation 
is  DE  (Fig.  10),  or  if  6  be  the 
angle  A  CE, 

z  =  r(i  —  cos  0). 
The  weight  of  the  cylinder  for  one  unit  of  length  is 

W  =  w'  .  nr\ 
and  that  of  the  displaced  water  is 

W=  w(f  arc  8  —  r*  sin  6  cos  6). 

Equating   the   values   of    W  and    W,  and   substituting   for 
sin  0  cos  6  its  equivalent  J  sin  20,  there  results 

w' 

2  arc  0  —  sin  26  =  2n  —  . 
w 

From  this  equation  0  is  to  be  found  by  trial  for  any  particular 


ART.  18.]  STABILITY  OF  FLOTATION.  33 

case,  and  then  z  is  known.     For  example,  if  w'  —  26.5  pounds 
per  cubic  foot,  and  r  =  12  inches, 

2  arc  8  —  sin  26  —  2.664  =  o. 

To  solve  this  equation,  assume  values  for  0,  until  finally  one  is 
found  that  satisfies  'it;  thus: 

For  6  —  83°,     2.897  —  0.242  —  2.664  =  —  0.009  J 
For  0  =  83i,     2.906  —  0.234  —  2.664  =  +  0.008. 


Therefore  0  lies  between  83°  and  83°  15'  ,  and  is  probably  about 
83°  8'.  Hence  the  depth  of  flotation  is  z  =  12(1  —0.120)  = 
10.6  inches. 

Prob.  24.    Show  that  the  depth  of  flotation  for  a  sphere 
whose   radius   is  r  is   the   real   root   of    the    cubic    equation 


Prob.  25.  A  rectangular  wooden  box  4.5  feet  long,  3  feet 
wide,  and  2.5  feet  deep,  inside  dimensions,  is  made  of  timber 
i|-  inches  thick,  which  weighs  3  pounds  per  foot  board  jmeasu  re. 
How  much  water  will  it  draw  when  a  weight  of  200  pounds  is 
placed  in  it  and  the  cover  nailed  on  ?  Ans.  0.46  feet. 


ARTICLE  18.  STABILITY  OF  FLOTATION. 

The  equilibrium  of  a  floating  body  is  stable  when  it  returns 
to  its  primitive  position  after  having  been  slightly  moved  there- 
from by  extraneous  forces,  it  is  indifferent  when  it  floats  in  any 
position,  and  it  is  unstable  when  the  slightest  force  causes  it  to 
leave  its  position  of  flotation.  For  instance,  a  short  cylinder 
with  its  axis  vertical  floats  in  stable  equilibrium,  but  a  long 
cylinder  in  this  position  is  unstable,  and  a  slight  force  causes  it 
to  fall  over  and  float  with  its  axis  horizontal  in  indifferent 
equilibrium. 


34  HYDROSTATICS.  [CHAP.  IL 

The  stability  depends  in  any  case  upon  the  relative  posi- 
tion of  the  centre  of  gravity  of  the  body  and  its  centre  of 
buoyancy,  the  latter  being  the  centre  of  gravity  of  the  dis- 
placed water.  Thus  in  Fig.  1 1  let  G  be  the  centre  of  gravity 

of  the  body  and  C  that  of  the 
centre  of  buoyancy  when  in  an 
upright  position.  Now  if  an  ex- 
traneous force  causes  the  body 
to  tip  into  the  position  shown, 
the  centre  of  gravity  remains 
at  G,  but  the  centre  of  buoy- 
ancy moves  to  D.  In  this  new 
FlG-  «•  position  of  the  body  it  is  acted 

upon  by  the  forces  W  and  W,  whose  lines  of  direction  pass 
through  G  and  D.  W  is  the  weight  of  the  body  and  W  the 
weight  of  the  displaced  water ;  and  as  these  are  equal,  they 
form  a  couple  which  tends  either  to  restore  the  body  to  the 
upright  position  or  to  cause  it  to  deviate  farther  from  that 
position.  Let  the  vertical  through  D  be  produced  to  meet  the 
centre  line  CG  in  M.  If  Mis  above  G  the  equilibrium  is  stable,, 
as  the  forces  W  and  W  tend  to  restore  it  to  its  primitive  posi- 
tion ;  if  M  coincides  with  G  the  equilibrium  is  indifferent ;  and 
if  MbQ  below  G  the  equilibrium  is  unstable. 

The  point  M  is  often  called  the  'metacentre,'  and  the 
theorem  may  be  stated  that  the  equilibrium  is  stable,  indifferent, 
or  unstable  according  as  the  metacentre  is  above,  coincident 
with,  or  below  the  centre  of  gravity  of  the  body.  The  measure 
of  the  stability  of  a  floating  body  is  the  moment  of  the  couple 
formed  by  the  forces  Wand  W.  But  the  line  GM is  propor- 
tional to  the  lever  arm  of  the  couple,  and  hence  the  quantity 
WX  GM  may  be  taken  as  a  measure  of  the  stability.  The 
stability,  therefore,  increases  with  the  weight  of  the  body,  and 
with  the  distance  of  the  metacentre  above  the  centre  of  gravity. 


ART,  18.]  STABILITY  OF  FLOTATION.  35 

To  ensure  a  high  degree  of  stability  the  centre  of  gravity  should 
be  as  low  as  possible. 

The  only  important  applications  of  these  principles  are  in 
connection  with  the  subject  of  naval  architecture,  and  in  general 
the  resulting  investigations  are  of  a  complex  character,  which 
can  only  be  solved  by  approximate  tentative  methods.  REED'S 
Treatise  on  the  Stability  of  Ships  (London,  1885)  is  a  large 
volume  entirely  devoted  to  this  topic. 

Prob.  26.  If  6  be  the  angle  of  inclination  to  the  vertical,  c 
the  distance  between  the  metacentre  and  centre  of  gravity, 
show  that  the  stability  of  flotation  can  be  measured  by  the 
quantity  We  sin  6. 


30  THEORETICAL  HYDRAULICS.  [CHAP.  III. 


CHAPTER  III. 

THEORETICAL  HYDRAULICS. 

ARTICLE  19.  VELOCITY  AND  DISCHARGE. 

If  a  vessel  or  pipe  be  constantly  full  of  water,  all  the  parti- 
cles of  which  move  with  the  same  uniform  velocity  v,  and  if  a 
be  the  area  of  its  cross-section,  the  quantity  of  water  which 
passes  any  section  per  second  is  equal  to  the  volume  of  a  prism 
whose  base  is  a  and  whose  length  is  v,  or 

q—av .     (6) 

If,  now,  the  vessel  varies  in  cross-section,  one  area  being  ay 
another  a^ ,  and  a  third  #2 ,  the  same  quantity  of  water  passes 
each  section  per  second  if  the  vessel  be  kept  constantly  full  ; 
hence  if  z>,  vl ,  and  v^  be  the  respective  velocities, 

q  =  av  —  a1^1  =  a^. 

The  velocities  of  flow  in  different  sections  of  a  pipe  or  vessel 
which  is  maintained  constantly  full  hence  vary  inversely  as 
the  areas  of  the  cross-sections. 

In  case  the  particles  or  filaments  move  with  different  veloci- 
ties in  different  parts  of  the  section,  the  quantity  may  be  still 


ART.  20.]          VELOCITY  OF  FLO W  FROM  ORIFICES.  37 

expressed  by  q  =  av,  provided  that  v  signifies  the  mean  velocity 
of  the  flow  ;  or 


a 


(6)' 


may  be  regarded  as  a  definition  of  the  term  mean  velocity. 

The  word  discharge  will  be,  used  to  denote  the  quantity  of 
water  flowing  per  second  from  a  pipe  or  orifice,  and  the  letter 
Q  will  designate  the  theoretic  discharge,  that  is,  the  discharge 
as  computed  by  the  methods  of  this  chapter,  where  resistances 
or  losses  due  to  friction,  contraction,  and  other  causes  are  not 
considered.  The  letter  Fwill  designate  the  theoretic  velocity, 
so  that  if  a  be  the  area  of  an  orifice,  or  the  cross-section  of  a  jet, 


is  the  formula  for  the  theoretic  discharge.  In  the  case  of  flow 
from  a  simple  orifice  the  area  a  is  found  by  the  measurement 
of  its  dimensions,  so  that  the  problem  of  finding  Q  is  reduced 
to  that  of  determining  V. 

Prob.  27.  A  pipe  constantly  filled  with  water  discharges 
0.43  cubic  feet  per  second.  Compute  the  mean  velocity  of  flow 
if  the  pipe  is  3  inches  in  diameter  ;  also  if  it  is  6  inches  in 
diameter. 

ARTICLE  20.  VELOCITY  OF  FLOW  FROM  ORIFICES. 

If  an  orifice  be  opened,  either  in  the  base  or  side  of  a  vessel 
containing  water,  it  flows  out  with  a  velocity  which  is  greater 
for  high  heads  of  water  than  for  low  heads.  The  theoretic 
velocity  of  flow  is  given  by  the  theorem  discovered  by  TORRI- 

CELLI.* 

The  theoretic  velocity  of  flow  at  the  orifice  is  the  same  as 
that  acquired  by  a  body  falling  freely  in  a  vacuum 
through  a  height  equal  to  the  head  of  water  on  the 
orifice. 

*Del  moto  dei  gravi  (Firenz,  1644). 


THEORETICAL   HYDRAULICS. 


[CHAP.  III. 


The  proof  of  this  rests  partly  on  observation.  Thus  if  a  vessel  be 

arranged,  as  in  Fig.  12,  so  that 
a  jet  of  water  from  an  orifice  is 
directed  vertically  upward,  it  is 
known  that  it  never  attains  to 
the  height  of  the  level  of  the 
water  in  the  vessel,  although 
under  favorable  conditions  it 
FIG.  12.  nearly  reaches  that  level.  It 

may  hence  be  inferred  that  the  jet  would  actually  rise  to  that 
height  were  it  not  for  the  resistance  of  the  air  and  the  friction 
of  the  edges  of  the  orifice.  Now,  since  the  velocity  of  impulse 
required  to  raise  a  body  vertically  to  a  certain  height  is  the 
same  as  that  acquired  by  it  in  falling  from  rest  through  that 
height,  it  is  regarded  as  established  that  the  velocity  at  the 
orifice  is  as  stated  in  the  theorem. 

The  following  proof  rests  on  the  law  of  conservation  of 
energy.  Let,  as  in  the  second  diagram  of  Fig.  12,  the  water 
surface  in  a  vessel  be  at  A  at  the  beginning  of  a  second  and  at 
Al  at  the  end  of  the  second.  Let  Z^be  the  weight  of  water 
between  the  planes  A  and  Alt  which  is  evidently  the  same  as 
that  which  flows  from  the  orifice  during  the  second.  Let  Wv  be 
the  weight  of  water  between  the  planes  Al  and  B,  and  h^  the 
height  of  its  centre  of  gravity  above  the  orifice.  Let  h  be  the 
height  of  A  above  the  orifice,  and  dh  the  distance  between  A 
and  Al  .  Then  at  the  beginning  of  the  second  the  water  in  the 
vessel  has  the  energy  Wfil  -f  W(h  —  ^#//).  If  Fbe  the  velocity 
of  flow  through  the  orifice,  the  same  water  at  the  end  of  the 


second  has  the  energy  WJi^  -J-  W  —  .     By  the  law  of  conserva- 

tion these  are  equal,  if  no  energy  has  been  dissipated  in  friction 
or  in  other  ways  ;  thus, 


*-***=-. 


ART.  20.]          VELOCITY  OF  FLOW  FROM  ORIFICES.  39 

Now  if  6/i  be  small  compared  with  h,  which  will  be  the  case 
when  A  is  large  compared  with  the  area  of  the  orifice,  this  gives 

v= 

which    is   the    same    as   for    a    body   falling    freely    through 
the  height  h  (Art.  6). 

The  theoretic  velocity  of  flow  from  any  orifice,  whether  its 
plane  be  horizontal,  vertical,  or  inclined,  is  thus  given  by 


(7) 


provided  the  orifice  be  small  compared  with  the  section  of  the 
reservoir.  The  theoretic  height  to  which  the  jet  will  rise  is 

F" 

k  =  -  .........      (7)' 

2g 

The  first  of  these  formulas  states  the  velocity  due  to  a  given 
head,  and  the  second  states  the  head  which  would  generate  a 
given  velocity.  The  term  "  velocity  head  "  will  be  generally 

V* 
used  to  designate  the  expression  —  ,  meaning  thereby  that  its 

o 

value  is  the  head  which  can  produce  the  velocity  V. 

Using   for  g  the  mean    value   32.16  feet  per   second  per 
second,  these  formulas  become 

V  =  8.02  VTi,        h  =  o.o  i  5  5  5  V\ 

from  which  the  following  tables  have  been  computed.  These 
are  mainly  intended  to  impress  upon  the  student  the  fact  that 
small  heads  produce  rapid  velocities,  but  they  may  also  prove 
serviceable  for  use  in  approximate  computations.  The  last 
columns  of  the  tables  give  multiples  of  the  numbers  8.02 
and  0.01555. 


40 


THE  ORE  TIC  A  L  H  YDRA  ULICS.  [CHAP.  III. 

TABLE    IV.    THEORETIC   VELOCITIES. 


Head. 
Feet. 

Velocity. 
Feet  per 
Second. 

Head. 
Feet. 

Velocity. 
Feet  per 
Second. 

Multiples  of 
8.01997. 

0.001 

0.254 

I 

8.02 

I 

8.02 

.002 

.358 

2 

n-33 

2 

16.04 

.003 

•  439 

3 

13.89 

3 

24.06 

.004 

•507 

4 

16.04 

4 

32.08 

•  005 

.567 

5 

17-93 

5 

40.  10 

.006 

.621 

6 

19.64 

6 

48.12 

.007 

.671 

7 

21.22 

7 

56.14 

.008 

.717 

8 

22.68 

8 

64.16 

.009 

.761 

9 

24.06 

9 

72.18 

TABLE  V.   VELOCITY   HEADS. 


Velocity. 
Feet  per 
Second. 

Head. 
Feet. 

Velocity. 
Feet  per 
Second. 

Head. 
Feet. 

Multiples  of 
0.015547. 

I 

0.0i6 

IO 

i-55 

I 

0.01555 

2 

.062 

20 

6.22 

2 

.03109 

3 

.140 

30 

13-99 

3 

.04664 

4 

.249 

40 

24.88 

4 

.06219 

5 

.389 

50 

38.86 

5 

•0/774 

6 

.560 

60 

55-97 

6 

.09328 

7 

.762 

70 

76.19 

7 

.10883   - 

8 

•995 

80 

99.51 

8 

.12438 

9 

1.260 

90 

125.95 

9 

.13992 

Prob.  28.  Find  the  theoretic  velocity  of  flow  from  an  orifice 
under  a  head  of  6  inches.     Find  the  velocity-head  of  a  stream 
O.I  feet  in  diameter  which  discharges  2.5  cubic  feet  per  minute- 
Ans.    V=  5.67  feet  per  second,  H  =  0.44  feet. 

ARTICLE  21.  HORIZONTAL  ORIFICES. 

Let  a  be  the  area  of  an  orifice  whose  plane  is  horizontal,  h 
the  head  of  water  upon  it,  and  Q  the  quantity  of  water  dis- 
charged per  second.  The  theoretic  discharge  is,  from  the  prin- 
ciples of  the  preceding  articles, 


Q  =  a  V2gk, 


(8) 


ART.  21.]  HORIZONTAL   ORIFICES.  41 

provided  that  the  area  of  the  orifice  be  small  compared  with 
the  cross-section  of  the  vessel.  If  a  is  in  square  feet  and  h  in 
feet,  Q  will  be  expressed  in  cubic  feet  per  second.  It  will  be 
seen  in  the  next  chapter  that  various  circumstances  materially 
modify  in  practice  the  results  as  obtained  from  this  formula. 

The  discharge  from  a  horizontal  orifice  is,  like  the  velocity, 
proportional  to  the  square  root  of  the  head.  Thus  with  the 
same  orifice  to  double  the  discharge  requires  the  head  to  be 
increased  fourfold.  The  head  which  will  produce  a  given  dis- 
charge is 


whence  the  head  varies  inversely  as  the  square  of  the  area  of 
the  orifice. 

Horizontal  orifices  are  but  little  used,  as  in  practice  it  is 
found  more  convenient  to  arrange  an  opening  in  the  side  of  a 
vessel  than  in  the  base.  The  above  formula  applies  approxi- 
mately to  a  vertical  orifice  if  h  be  taken  as  the  head  on  its 
centre  of  gravity. 

The  discharge  is  theoretically  independent  of  the  shape  of 
the  orifice,  so  that  orifices  of  different  forms  with  equal  areas 
give  the  same  value  of  Q.  For  a  circle  whose  diameter  is  d, 


Q  = 
For  a  rectangle  whose  sides  are  b  and  d, 


and  similarly  for  other  forms  a  is  to  be  inserted  in  terms  of  the 
linear  dimensions,  which  must  be  numerically  expressed  in  the 
same  unit  as  g. 

Prob.  29     Compute  the  theoretic  discharge  from  an  orifice 
one  inch  in  diameter  under  a  head  of  1.5  feet. 


THE  ORE  TIC  A  L  H  YDRA  ULICS. 


[CHAP.  III. 


charge  is  a  V  or  a  V2gk. 


ARTICLE  22.  RECTANGULAR  VERTICAL  ORIFICES. 

If  the  size  of  an  orifice  in  the  side  of  a  vessel  be  small  com- 
pared with  the  head,  then  the  mean  theoretic  velocity  of  the 
outflowing  water  may  be  taken  as  V2gh,  where  Ji  is  the  head  on 
the  centre  of  the  orifice,  and  consequently  the  theoretic  dis- 

Strictly,  however,  the  head,  and  hence 
the  velocity,  is  different  in  dif- 
ferent parts  of  an  orifice  whose 
plane  is  vertical. 

A  rectangular  orifice  with 
two  edges  parallel  to  the  water 
surface  is  the  most  important 
case.  Let  b  be  its  breadth,  h^ 
the  head  of  water  on  its  upper 
edge,  and  /*3  the  head  on  its 
lower  edge,  so  that  h^  —  hl  is  its  depth.  Let  any  elementary 
strip  whose  area  is  b .  Sy  be  drawn  at  a  depth  y  below  the  water 
level.  The  velocity  of  flow  through  this  elementary  strip  is,  as 
shown  in  Art.  20, 


FIG.  13. 


and  the  discharge  per  second  through  it  is 


dQ  =  bdy  V2gy. 

The  total  discharge  through  the  orifice  is  obtained  by  integrat- 
ing this  expression  between  the  limits  7z,  and  hz  ,  which  gives 

Q  =  lb  V&W  -  A,t)  .......    (9) 

In  case  the  top  edge  of  the  orifice  is  at  or  above  the  level  of 
the  water,  h^  =  o,  and  then  if  the  head  h^  be  denoted  by  H, 
the  discharge  is 

=  \a  V~2gH,   .     .     (9)' 


which  is  the  basis  of  all  formulas  for  weir  measurement. 


ART.  22.]  RECTANGULAR    VERTICAL   ORIFICES.  43 

To  ascertain  the  error  caused  by  using  the  formula  (8)  in- 
stead of  (9)  for  a  rectangular  lateral  orifice,  let  h  be  the  head  on 
its  centre  of  gravity,  and  d  be  its  vertical  depth,  //3  —  /^  .  Then 
from  (8) 

Q  =  bdV2gh. 

Now  in  (9)  let  h^  =  h  -f-  \d,  and  //,  =  h—-\d\  then  developing 
by  the  binomial  formula, 


and  (9)  becomes 

i  d 


,  y  __  /          i  i  i  \ 

fi  ==  V^(i  -  ^g/7  -  —  g^  -  ^  ?7  -  etc.). 


Therefore  the  discharge  obtained  by  using  (8)  is  always  too 
great.  The  true  theoretic  discharge,  from  the  formula  just 
deduced,  is: 

If     h  =    d,  Q  =  0.989  bd  V~2gh\ 

li     k  =  2d,  Q  =  0.997  bd 


If     //  =  3</,  Q  =  0.999  bd  V~2gk. 

The  error  of  the  formula  Q  =  bd  ^2gh  is  thus  seen  to  be  i.i 
per  cent  when  //  =  d,  only  0.3  per  cent  when  h  =  2d,  and  only 
about  o.i  per  cent  when  h  =  ^d.  Accordingly,  if  the  head  on 
the  centre  of  the  orifice  is  greater  than  two  or  three  times  the 
vertical  depth  of  the  orifice,  the  approximate  formula  (8)  is 
generally  used  instead  of  the  exact  formula  (9),  since  the  slight 
error  thus  introduced  is  of  no  practical  importance. 

Prob.  30.  Compute  the  theoretic  discharge  from  a  rectan- 
gular orifice  0.5  feet  wide  and  0.25  feet  high  when  the  head  on 
the  top  of  the  orifice  is  0.375  feet. 

Ans.  Q  =  0.707  cubic  feet  per  second. 


44 


THEORE  TIC  A  L   H  YDRA  ULICS. 


[CHAP.  III. 


FIG.  14. 


ARTICLE  23.  TRIANGULAR  VERTICAL  ORIFICES. 
Triangular   vertical    orifices   arc   sometimes  used    for   the 

measurement  of  water,  the  arrangement   being   as   shown  in 

Fig.  14.  Let  b  be  the  width  of 
the  orifice  at  the  water  level, 
and  //  the  head  of  water  on 
the  vertex.  Let  an  elementary 
strip  whose  depth  is  dy  be 
drawn  at  a  distance  7  below  the 

water  level.     From  similar  triangles  the  length  of  this  strip  is 

-jj(H—y\  and  the  elementary  discharge  then  is 


-  y)dy  V^gy  =  -- 


The  integration  of  this  between  the  limits  o  and  H  gives 

Q  =  -hb  1/2JH*  =  -^bHV^H. 

If  the  sides  of  the  triangle  are  equally  inclined  to  the  vertical, 
as  should  be  the  case  in  practice,  and  if  this  angle  be  a,  b  may 
be  expressed  in  terms  of  a  and  //,  so  that  the  equation  be- 
comes 

Q  =  TV  tan  a  .  IT  V~2gH=  -ft-  tan  a  .  Vzg  .  H*. 

The  discharge  is  thus  equal  to  a  constant  multiplied  by  the 
2\  power  of  the  measured  depth. 

If  the  orifice  be  a  trapezoid  whose  upper  base  is  b,  lower 
base  bf,  and  altitude  d,  the  discharge  is  found  by  integrating 
the  above  differential  expression  between  the  limits  o  and  d, 
and  then  substituting  for  H  its  value  in  terms  of  d,  b,  and  b'  r 

namely,  H  =  -7  -  -77  .     The  theoretic  discharge  then  is 


ART.  24.] 


CIRCULAR    VERTICAL   ORIFICES. 


45 


If  in  this  b'  equals  b  it  becomes  the  same  as  the  formula  for  a 
rectangular  orifice,  while  if  b'  equals  o  it  gives  the  same  result 
as  found  above  for  the  triangle. 

Prob.  31.  Prove  that  the  theoretic  discharge  from  a  lateral 
triangular  orifice  whose  base  is  horizontal  and  whose  vertex  is 
in  the  water  level  is  Q  =  \bd  V2gd,  where  b  is  the  base  and  d\s 
the  altitude. 


ARTICLE  24.  CIRCULAR  VERTICAL  ORIFICES. 

To  determine  the  theoretic  discharge  through  a  circular 
orifice  whose  plane  is  vertical,  let  h  be  the  head  on  its  centre, 
and  r  its  radius.     Let   an  elementary 
strip  be  drawn  at  a  distance  y  above 
the   centre  ;      the   length    of    this    is 
2  Vr*  —  y  ,    its    area    is   2$y  Vr*  —  y*, 
and  the  head  upon  it  is  h  —  y.     Then 
the  theoretic  discharge   through   this 
strip  is  _     _ 

$Q  =  2dy  Vr*  —  y*  V  '  2&(h  —  y).  FlG-  «s. 

To  integrate  this  expand  (h  —  yfi  by  the  binomial    formula. 
Then  it  may  be  written 


Each  term  of  this  expression  is  now  integrable,  and  taking  the 
limits  of  y  as  -f-  r  and  —  r  the  entire  circle  is  covered,  and 


which  gives  the  theoretic  discharge  per  second  for  any  values 
of  r  and  h. 


46  THEORETICAL   HYDRAULICS.  [CHAP.  III. 

The  approximate  formula  (8)  applied  to  this  case  gives  for 
the  discharge  nr*  V2gh,  which  is  always  greater  than  the  true 
discharge  ;  thus  from  (10), 

If  h  =  2r,  Q  =  0.992  itr* 
If  h  =  $r,  Q  =  0.996  Trr2 
If  h  =  4r,  Q  =  0.998  nr* 

Hence  the  error  in  the  use  of  (8)  is  only  0.4  per  cent  when 
k  =  $r,  and  only  0.2  per  cent  when  h  —  47*.  In  general  the 
approximate  formula  may  be  used  whenever  the  head  on  the 
centre  of  the  circle  is  greater  than  four  or  five  times  its  radius. 

Prob.  32.  Compute  the  theoretic  discharge  from  a  circle  of 
one  inch  diameter  when  the  head  on  its  centre  is  0.5  feet. 


ARTICLE  25.  INFLUENCE  OF  VELOCITY  OF  APPROACH. 

Thus  far,  in  the  determination  of  the  theoretic  velocity  and 
discharge  from  an  orifice,  the  head  has  been  regarded  as  con- 
stant. But  the  head  can  only  be  maintained  constant  by  an 
inflow  of  water,  and  this  modifies  the  theoretic  velocity.  Let 
a  be  the  area  of  the  orifice,  and  A  that  of  the  horizontal  cross- 
section  of  the  reservoir;  let  Fbe  the  theoretic  velocity  of  flow 
through  #,  and  v  the  vertical  velocity  of  inflow  through  the 
section  A.  The  energy  of  Impounds  of  water  as  it  flows  from 

J7a 
the  orifice  is  W  -—  ,  and  this  is  equal  to  the  energy  Wk  stored 

up  in  the  fall  plus  the  energy  W  —  of  the  inflowing  water,  or 

<*> 


W—  =  Wh  +  W—. 
tg  zg 

Now  the  quantity  of  water  which  flows  through  the  section  a 


ART.  25.]     INFLUENCE  OF   VELOCITY  OF  APPROACH.  47 

in  a  unit  of  time  is  the  same  as  that  passing  through  the  area 
A  in  the  same  time,  or  (Art.  19) 

aV=  Av,     whence     v  =  —  V. 

A 

Inserting  this  value  of  v  in  the  equation  of  energy,  and  solving 
for  F,  gives  the  result 


v7.  •% 


~  ~  (II) 


which  is  always  greater  than  the  value  V2gh. 

The  influence  of   a  constantly  maintained  head  on  the  ve- 
locity of  flow  at  the  orifice  can  now  be  ascertained  by  assign- 

ing values  to  the  ratio  —  t  thus  : 
A 

If  a  —       A,  V=  oo  ; 

If  a  =  \  A,  V—  1.342 

If  a  =  \A,  V=  1.154 

If  a  =  $A,  V—  1.061 

If  a—\A,  V  =  1.021 

If  a  =  TV4,  F  =  1.005 


It  is  here  indicated  that  the  common  formula  (8)  is  in  error 
2.1  per  cent  when  a  =  %A,  if  the  head  be  maintained  constant 
by  a  uniform  vertical  inflow  at  the  water  surface,  and  0.5  per 
cent  when  a  =  -faA.  Practically,  if  the  area  of  the  orifice  be 
less  than  one-twentieth  of  the  cross-section  of  the  vessel,  the 
error  in  using  the  formula  F—  V2gk  is  too  small  to  be  noticed 
even  in  the  most  precise  experiments,  and  fortunately  most 
orifices  are  smaller  in  relative  size  than  this. 

A  more  common  case  is  that  where  the  reservoir  is  of  large 


48 


THE  ORE  TIC  A  L   PI  YDRA  ULICS. 


[CHAP.  III. 


horizontal  and  small  vertical  cross-section,  and  where  the  water 

approaches  the  orifice  with  a 
horizontal  velocity,  as  in  a 
canal  or  trough.  Here  let  A 
be  the  area  of  the  vertical 
cross-section  of  the  vessel,  a 
the  area  of  the  orifice,  and  h 
FIG.  16.  the  head  upon  its  centre. 

Then  if  h  be  large  compared  with  the  depth   of   the  orifice, 

exactly  the  same  reasoning  applies  as  before,  and  the  theoretic 

velocity  of  flow  is 


/     Zg* 


If,  however,  h  be  small,  let  hl  and  htl  be  the  heads  on  the  upper 
and  lower  edges  of  the  orifice,  which  is  taken  as  rectangular. 
Then,  using  the  same  reasoning  as  above,  the  velocity  of  flow 
at  any  depth  y  is  given  by 


where  v  is  the  constant  velocity  of  approach  through  the  area 
A.  The  discharge  through  a  strip  of  the  length  b  and  depth 
dy  (Art.  20)  then  is 


and,  by  integration  between  the  limits  /^  and  h^  ,  the  theoretic 
discharge  per  second  from  the  orifice  is 


In  this  case,  particularly  when  kl  =  o,  the  velocity  of  approach 
may  exercise  a  marked  influence  on  the  discharge. 


ART.  26.]  FLOW   UNDER  PRESSURE.  49 

Prob.  33.  In  the  case  of  horizontal  approach,  as  seen  in 
Fig.  1 6,  let  b  =  4  feet,  7/2  =  0.8  feet,  /^  =  o,  and  v  =  2.5  feet 
per  second.  Compute  the  theoretic  discharge  :  first,  neglecting 
v ;  and  second,  regarding  v. 


ARTICLE  26.  FLOW  UNDER  PRESSURE. 

The  level  of  water  in  the  reservoir  and  the  orifice  of  out- 
flow have  been  thus  far  regarded  as  subjected  to  no  pressure, 
or  at  least  only  to  the  pressure  of  the  atmosphere  which  acts 
upon  both  with  the  same  mean  force  of  14.7  pounds  per  square 
inch  (since  the  head  h  is  rarely  or  never  so  great  that  a 
sensible  variation  in  atmospheric  pressure  can  be  detected 
between  the  orifice  and  the  water  level).  But  the  upper  level 
of  the  water  may  be  subject  to  the  pressure  of  steam  or  to  the 
pressure  due  to  a  heavy  weight  or  to  a  piston.  The  orifice 
may  also  be  under  a  pressure  greater  or  less  than  that  of  the 
atmosphere.  It  is  required  to  determine  the  velocity  of  flow 
from  the  orifice  under  these  conditions. 

First,  suppose  that  the  surface  of  the  water  in  the  vessel  or 
reservoir  is  subjected  to  the  uniform  pressure  of  p0  pounds  per 
square  foot  above  the  atmospheric  pressure,  while  the  pressure 
at  the  orifice  is  the  same  as  that  of  the  atmosphere.  Let  h  be 
the  depth  of  water  on  the  orifice.  The  velocity  of  flow  V  is 
greater  than  V2gk  on  account  of  the  pressure  /„ ,  and  it  is 
evidently  the  same  as  that  from  a  column  of  water  whose 
height  is  such  as  to  produce  the  same  pressure  at  the  orifice. 
The  total  unit-pressure  at  the  depth  of  the  orifice  is 

p  =  wh-\-pOJ 

and  from  (i)  the  head  of  water  which  would  produce  this  pres- 
sure is 

I -•*+.&, 

w  w 


50  THEORETICAL   HYDRAULICS.  FCHAP.  III. 

Accordingly  the  velocity  of  flow  from  the  orifice  is 


or,  if  h0  denote  the  head  corresponding  to  the  pressure  A 
V  = 


The  general  formula  (6)  thus  applies  to  any  small  orifice,  if  h 
be  the  head  corresponding  to  the  static  pressure  at  the  orifice. 

Secondly,  suppose  that  the  surface  of  the  water  in  the 
vessel  is  subjected  to  the  unit-pressure  A  >  while  the  orifice  is 
under  the  external  unit-pressure  pr  Let  h  be  the  head  of 
actual  water  on  the  orifice,  /i0  the  head  of  water  which  will 
produce  the  pressure  A  >  and  /^  the  head  which  will  produce  A- 
The  velocity  of  flow  at  the  orifice  is  then  the  same  as  if  the 
orifice  were  under  a  head  h  -f-  h0  —  /^ ,  or 


//0-/0,    .....    (12) 
in  which  the  values  of  /i0  and  /*,  are 


w 


Usually  A  and/,  are  given  in  pounds  per  square  inch,  while  h^ 
and  h^  are  required  in  feet  ;  then  (Art.  9) 

/i0=  2.304  A,         ^r=  2.304  A- 

The  values  of  pQ  and  A  may  be  absolute  pressures,  or  merely 
pressures  above  the  atmosphere.  In  the  latter  case  A  ma7 
sometimes  be  negative,  as  in  the  discharge  of  water  into  a 
condenser. 

As  an  illustration  of  these  principles  let  a  cylindrical  reser- 


ART.  26.] 


FLO  W   UNDER   PRESSURE. 


,w 


voir,  Fig.  17,  be  2  feet  in  diameter,  and  upon  the  surface  of  the 
water  let  there  be  a  tightly  fitting 
piston  which  with  the  load  W 
weighs  3000  pounds.  At  the 
depth  8  feet  below  the  water 
level  are  three  small  orifices :  one 
at  A,  upon  which  there  is  an  ex- 
terior head  of  water  of  3  feet;  one 
not  shown  in  the  figure,  which 
discharges  directly  into  the  at" 
mosphere ;  and  one  at  C,  where  the  discharge  is  into  a  vessel  in 
which  the  tension  of  the  air  is  only  ro  pounds  per  square  inch. 
It  is  required  to  determine  the  velocity  of  efflux  from  each 
orifice.  The  head  hQ  corresponding  to  the  pressure  on  the 
upper  water  surface  is 

3000 


FIG.  17. 


W 


=2 7- —  =  15.28  feet. 

3.1416  x  62.5 


The  head  /^  is  3  feet  for  the  first  orifice,  o  for  the  second,  and 
—  2.304(14.7—  10)  —  —  10.83  feet  f°r  the  third.  The  three 
theoretic  velocities  of  outflow  then  are  : 

V  —  8.02  4/8+15.28—  3  =  36.1  feet  per  second  ; 
V  —  8.02  V%+  15.28  -  ~~b~  =  38.7  feet  per  second  ; 
V  =  8.02  VS+  15.28+  10.83"=  46.8  feet  per  second. 

In  the  case  of  discharge  from  an  orifice  under  water,  as  at 
A  in  Fig.  17,  the  value  of  h  —  hl  is  the  same  wherever  the 
orifice  be  placed  below  the  lower  level,  and  hence  the  velocity 
depends  upon  the  difference  of  level  of  the  two  water  surfaces, 
and  not  upon  the  depth  of  the  orifice. 


The  velocity  of  flow  of  oil  or  mercury  under  pressure  is  to 
be   determined  in  the  same  manner  as  water,  by  finding  the 


52  THEORETICAL  HYDRAULICS.  [CHAP.  III. 

heads  which  will  produce  the  given  pressure.  Thus  in  the  pre- 
ceding numerical  example,  if  the  liquid  be  mercury,  whose 
weight  per  cubic  foot  is  850  pounds,  the  head  of  mercury  cor- 
responding to  the  pressure  of  the  piston  is 

3000 

7/0  — -p. —  =  1. 1 2  feet, 

3.1416  X  850 

and,  accordingly,  for  discharge  into  the  atmosphere  at  the 
depth  h  =  8  feet  the  velocity  is 

V  =  8.02  1/8  +  1. 1 2  =  24.2  feet  per  second, 

while  for  water  the  velocity  was  38.7  feet  per  second.  The 
general  formula  (6)  is  applicable  to  all  cases  of  the  flow  of 

P 
liquids  from  a  small  orifice,  if  for  h  its  value  —-be  substituted, 

where/  is  the  resultant  unit-pressure  at  the  depth  of  the  orifice, 
and  w  is  the  weight  of  a  cubic  unit  of  the  liquid. 

Prpb.  34.  Water  under  a  head  of  230  feet  flows  into  a  boiler 
whose  gauge  reads  45  pounds  per  square  inch.  Find  the  ve- 
locity of  the  inflowing  water. 

Prob.  35.  The  pressure  in  a  boiler  is  60  pounds  per  square 
inch  above  the  atmosphere.  Compute  the  theoretic  velocity 
of  flow  from  a  small  orifice  one  foot  below  the  water  level. 


ARTICLE  27.  PRESSURE-HEAD  AND  VELOCITY-HEAD. 

When  a  vessel  is  filled  with  water  at  rest  the  pressure  at 
any  point  depends  only  upon  the  head  of  water  above  that 
point  (Art.  9).  But  when  the  water  is  in  motion  it  is  a  fact  of 
observation  that  the  pressure  becomes  less  than  that  due  to 
the  head.  The  actual  pressure  in  any  event  may  be  measured 
by  the  height  of  a  column  of  water.  Thus  if  the  water  be  at 


ART.  27.]        PRESSURE-HEAD  AND    VELOCITY-HEAD. 


53 


rest  in  the  case  shown  in  Fig.  18,  and  small  tubes  be  inserted 
at  A,  B,  and  C,  the  water  will 

rise  in  each  tube  to  the  same    

height  as  that  of  the  water  f 


level  in  the  reservoir,  and  the   1^.~ 

pressures  at  A,  B,  and  C  will  X— 

be  those  due  to  the  heads  Aa, ~A-~-~^^~— ~- 

Bb,  and  Cc.     But  if  an  orifice  

be  opened,  as  seen  near  C,  the  FlG-  *z. 

water  levels  in  the  tubes  sink  to  the  points  alt  blt  and  cl ;  that 
is,  the  pressures  at  A,  B,  and  C  are  reduced  to  those  due  to 
the  heads  Aa, ,  Bbl ,  and  Ccl . 


Let  h  be  the  head  of  water  on  any  point,  or  the  depth  of 
that  point  below  the  free  water  level.  Let  h^  be  the  head 
due  to  the  actual  pressure  of  the  water  at  that  point,  or  the 

v* 
pressure-head.     Let  —  be  the  head  due  to  the  actual  velocity 

of  the  water  at  that  point,  or  the  velocity-head.     Then 


— 
2g 


d3) 


or,  in  the  form  of  a  theorem : 


The  pressure-head  plus  the  velocity-head  is  equal  to  the 
total  hydrostatic  head, 

In    order  to  prove  this  let  W  be  the  weight  of  water  which 

v* 
passes  the  section  per  second  ;  then  W  —  is  the  energy  which 

o 

it  possesses.     The  total  theoretic  energy  of  this  water  is  Wh, 
and  if  there  be  no  losses  of  energy  the  remaining  energy  is 


W  \h  --  1  ,  which  is  to  be  equated  to  Whl  ,  which  represents 
the   potential   energy  still  existing  in  the  form  of  pressure. 


54  THEORETICAL  HYDRAULICS.  [CHAP.  III. 

v"* 
Hence  h — h^ , 

whence  the  theorem  follows  as  stated.      In  Fig.  18  aa^  is  the 
velocity-head  for  the  section  A,  while  Aal  is  the  pressure-head. 

Another  method  of  proof  is  to  consider  the  section  at  A  as 
an  orifice  through  which  the  flow  occurs  under  a  head  h  —  /z, , 
where  h^  is  the  head  caused  by  the  back  pressure  p^ .  Then, 
from  the  last  article, 


V* 

from  which  —  =  h  —  h^ ,  which  also  agrees  with  the  theorem. 

The  pressure-head  Aa^  at  A  hence  decreases  when  the  ve- 
locity of  the  water  at  A  increases,  and  the  same  is  true  for  any 
other  section  as  B.  Let  v  and  v'  be  the  velocities  at  A  and  B ; 
then,  since  the  same  quantity  of  water  passes  each  section  per 
second,  the  relation  Av  =  Bv'  must  be  fulfilled.  Hence  if  B 
be  greater  than  A  the  velocity  v  is  greater  than  v' ,  and  the 
pressure-head  at  B  will  be  greater  than  at  A.  To  illustrate  :  let 
the  depths  of  A  and  B  be  6  and  5  feet  respectively  below  the 
water  level,  and  the  corresponding  cross-sections  be  1.2  and  2.4 
square  feet.  Let  the  quantity  of  water  discharged  by  the 
orifice  near  C  be  14.4  cubic  feet  per  second.  Then  the  velocity 
at  A  is 

14.4 
v  = =12  feet  per  second, 

which  corresponds  to  a  velocity-head  of 

—  =  0.015 55^a  =  2.24  feet ; 

and  accordingly  the  pressure-head  Aav  is 

//,  =  6.0  —  2.24  =  3.76  feet. 


ART.  27.]       PRESSURE-HEAD   AND    VELOCITY  HEAD. 


55 


Proceeding  in  the  same  way  for  B,  the  velocity  is  found  to  be 
6  feet  per  second,  the  velocity-head  0.56  feet,  and  finally  the 
pressure-head  is  5.0  —  0.56  =  4.44  feet.  The  hydrostatic  head 
at  A  is  thus  diminished  by  the  velocity-head  aal  =  2.24  feet, 
while  at  B  it  is  diminished  by  the  smaller  'amount  bbl  =  0.56 
feet.  When  the  water  was  at  rest  the  pressures  were  : 

At  A,     p  =  0.434  X  6  =  2.60  pounds  per  square  inch ; 
At  By     p  =  0.434  X  5  =  2.17  pounds  per  square  inch. 

But  as  soon  as  the  flow  from  the  orifice  began  the  pressures 
became : 

At  A,     p   =  0.434  X  3.76  =  1.63  pounds  per  square  inch  ; 
At  B,     p   =  0.434  X  4-44  =  1.93  pounds  per  square  inch. 


A  negative  pressure  may  occur  if  the  velocity-head  becomes 
greater  than  the  hydrostatic  head;  for  since. hv  -\ equals  //, 

the  value  of  hl  is  negative  when  —  exceeds  h.      A  case  in 

which  this  may  occur  is  shown  in  Fig.  19,  where  the  section  at 

v* 
A    is   so   small   that  —   becomes 


larger  than  k,  so  that  if  a  tube  be 
inserted  no  water  runs  out,  but  if 
the  tube  be  carried  downward  into 
a  vessel  of  water  there  will  be 
lifted  a  column  CD  whose  height 
is  that  of  the  negative  pressure- 
head  hl.  For  example,  let  the 
cross-section  of  A  be  0.4  square 


FIG.  19. 


feet,  and  its  head  h  be  4.1  feet,  while  8  cubic  feet  per  second 
are  discharged  from  the  orifice  below.  Then  the  velocity  at  A 
is  20  feet  per  second,  and  the  corresponding  velocity-head  is 


56  THEORETICAL   HYDRAULICS.  [CHAP.  III. 

6.22  feet.     The  pressure-head  at  A  then  is,  from  (13), 

//,  =  4.1  —  6.22  =  —  2.12  feet, 

and  accordingly  there  exists  at  A  an  inward,  or  negative 
pressure, 

/,  =  —  2.12  X  0.434  —  —  °-92  pounds  per  square  inch. 

This  negative  pressure  will  sustain  a  column  of  water  CD 
whose  height  is  2.12  feet.  If  the  small  vessel  be  placed  so  that 
its  water  level  is  less  than  2.12  feet  below,  water  will  be  con- 
stantly drawn  from  the  smaller  to  the  larger  vessel.  This  is 
the  principle  of  the  action  of  the  injector-pump. 

Prob.  36.  The  hydrostatic  pressure  in  a  pipe  is  80  pounds 
per  square  inch.  What  velocity  must  the  water  have  to  reduce 
this  to  50  pounds  per  square  inch  ? 

ARTICLE  28.  TIME  OF  EMPTYING  A  VESSEL. 

Let  the  depth  of  water  in  a  vessel  be  H\  it  is  required  to 
determine  the  time  of  emptying  it  through  a  small  orifice  in 
the  base  whose  area  is  a.     Let  Y  be  the  area 
of  the  water  surface  when  the  depth  of  water 
is  y\  let  dt  be  the  time  during  which  the  water 
level   falls  the  distance  6y.     During  this  time 
the  quantity  of  water  Y6y  passes  through  the 
FIG.  20.  orifice.    But  the  discharge  in  one  second  under 

the  constant  head  y  is  a  V2gy,  and  hence  the  discharge  in  the 
time  dt  is  a&t  V2gy.  Equating  these  two  expressions,  there  is 
found  the  relation 


. 
a  V2gy 

The  time  of  emptying  the  vessel  is  now  found  by  inserting  for 
Y  its  value  in  terms  of  y,  and  then  integrating  between  the 
limits  H  and  o. 


ART.  28.]  TIME   OF  EMPTYING  A    VESSEL.  57 

For  a  cylinder  or  prism  the  cross-section  Y  has  the  constant 
value  A,  and  the  formula  becomes 

Ay  -  *6y 
dt  =  —--=^-> 
aV  2g 

the  integration  of  which  gives 

2A   /~H          2AH 


t  — 


a  V2g         a  \/2gH 

as  the  theoretic  time  of  emptying  the  vessel.  If  the  head  were 
maintained  constant  the  uniform  discharge  per  second  would 
be  a  V2gH,  and  the  time  of  discharging  a  quantity  equal  to 
the  capacity  of  the  vessel  is  AH  divided  by  a  \/2gH,  which  is 
one  half  of  the  time  required  to  empty  it. 

To  find  the  time   of   emptying  a  hemispherical   bowl  of 
radius  r,  let  x  be  the  radius  of  the  cross-section  Y\  then 


=  2ry  —  /; 


Y=n(2ry-y*). 
The  equation  for  6t  then  becomes 


and  by  integration  between  the  limits  r  and  o 


which  is  the  theoretic  time  required  to  empty  the  hemisphere. 

The  only  important  application  of  these  principles  is  in  the 
case  of  the  right  prism  or  cylinder,  and  the  formula  for  this 
is  materially  modified  in  practice,  as  will  be  seen  in  the  next 
chapter.  It  is  more  frequently  required  to  determine  the 
time  during  which  the  water  level  will  descend  from  the 


58  THEORETICAL  HYDRAULICS.  [CHAP.  III. 

height  H  to  another  height  h.     This  is  found  by  integrating 
between  the  limits  H  and  h  ;  thus,  for  the  prismatic  vessel, 


which  gives  the  theoretic  time  of  descent  in  seconds. 

Prob.  37.  A  sphere  is  filled  with  water.  Find  the  time  of 
emptying  it  through  a  small  orifice  at  its  lowest  point. 

Prob.  38.  A  conical  vessel  whose  altitude  is  H,  and  whose 
base  has  the  radius  r,  is  placed  with  its  axis  vertical,  and 
emptied  through  a  small  orifice  in  its  base.  Prove  that  the 

i67r 
theoretic  time  is 


ARTICLE  29.  FLOW  FROM  A  REVOLVING  VESSEL. 

The  water  in  a  vessel  at  rest  is  acted  upon  only  by  the 
force  of  gravity,  and  hence  its  surface  is  a  horizontal  plane  ;  but 
the  water  in  a  revolving  vessel  is  acted  upon  by  a  centrifugal 
force  as  well  as  by  gravity,  so  that  its  surface  assumes  a  curved 
shape.  The  simplest  case  is  that  of  a  vessel  revolving  with 
uniform  velocity  about  a  vertical  axis,  and  it  will  be  shown 
that  here  the  water  surface  forms  a  paraboloid  whose  axis 
coincides  with  that  about  which  it  revolves.  Fig.  21  repre- 
sents such  a  case,  NT  being  the  vertical  axis. 

Let  J/be  any  point  on  the  surface  whose  co-ordinates  ON 
and  NM  are  y  and  x.  Let  W  be  the 
weight  of  a  particle  at  M,  whose  intensity 
is  represented  by  MG  ;  this  particle  in 
consequence  of  its  velocity  of  revolution 
u  is  acted  upon  also  by  a  centrifugal  force 

W    if 

MC  whose  value*  is  ---  —  The  resultant 
__  g     x  _ 

*  See  WOOD'S  Elementary  Mechanics,  p.  226. 


ART.  29.]  FLOW  FROM  A   REVOLVING    VESSEL,  S9 

MR  of  the  weight  and  centrifugal  force  must  be  normal  to  the 
tangent  MS  at  M,  as  the  condition  of  equilibrium.  The  angle 
NMS  is  hence  equal  to  RMG,  and  accordingly 


MG      gx 
* 
But  the  tangent  of  this  angle  is  the  first  derivative  of  y  with 

reference  to  x.  Further,  the  value  of  u  varies  directly  with 
x,  so  that  u  =  GOX  if  GO  be  the  angular  velocity,  that  is,  the 
velocity  at  the  distance  unity  from  the  axis.  Accordingly, 

6y  _  u*  _  ca1 

6x      gx       g 

is  the  differential  equation  of  the  curve,  and  by  integration 


which  is  the  equation  of  a  common  parabola.  Therefore  the 
surface  is  a  paraboloid.  Since  cox  is  the  velocity  u  at  the  point 
Mt  this  equation  may  be  written 


which  shows  that  the  ordinate  y  is  the  head  due  to  the  velocity 
of  revolution. 

If  h  be  the  head  OT  at  the  axis,  the  velocity  of  efflux 
from  a  small  orifice  at  T  is  V2gh.  But  for  an  orifice  at  U  the 
velocity  is  due  to  the  head  MU,  and 

MU— 


The  theoretic  velocity  of  flow  from  U  therefore  is 


V=  V2g(h  +  y)  =  V2gh  +  u\  .    .     .     .     (IS) 

where  u  is  the  velocity  of  revolution  of  the  point  U  or  M. 
This  formula  is  a  very  important  one  in  the  discussion  of  cer- 
tain hydraulic  motors. 


60  THE  ORE  TIC  A  L  H  YDRA  ULICS.  [CHAP.  1  1  L 

To.  determine  the  velocity  u  of  a  point  at  the  distance  x 
from  the  axis  of  revolution  it  is  only  necessary  to  count  the 
number  of  revolutions  made  per  second.  If  n  be  this  number, 

u  =  2nx  .  n  ; 

or,  in  another  form,  since  2nn  is  the  velocity  at  the  distance 
unity  from  the  axis, 

oo  =  2nn       and       u  =  GOX. 

As  an  example  of  the  application  of  these  principles,  let 
there  be  a  cylindrical  vessel  which  is  2  ^feet  in  diameter  and 
3  feet  deep,  and  which  is  one  half  full  of  water.  It  is  required 
to  find  the  number  of  revolutions  per  second  about  its  axis 
which  will  cause  the  water  to  begin  to  overflow  around  the 
upper  edge.  The  volume  of  a  paraboloid  being  one-half  of 
its  circumscribing  cylinder,  the  vertex  of  the  paraboloid  at  the 
moment  of  overflow  will  coincide  with  the  centre  of  the  base 
of  the  vessel,  and  hence  the  value  of  y  for  the  upper  edge  is 
3  feet.  Accordingly, 


whence  GO  =  13.89,  and  then 


which  is  the  number  of  revolutions  per  second.  If  the  vessel 
were  three-fourths  full  of  water,  the  volume  of  the  paraboloid 
at  the  moment  of  overflow  would  be  one-fourth  that  of  the 
cylinder,  and  the  value  of  y  for  the  upper  edge  would  be  one- 
half  the  altitude  of  the  cylinder,  or  1.5  feet.  Hence  GO  is  found 
to  be  9.82,  whence  the  number  of  revolutions  per  second  is 
about  1.56. 

Prob.  39.  A  cylindrical  vessel  is  3  feet  in  diameter.     How 
many  revolutions  per  minute  must  be  made  about  its  vertical 


ART.  30.]  THE  PATH  OF  A  JET.  6 1 

axis  in  order  that  the  velocity  of  the  outer  surface  may  be  50 
feet  per  second  ? 

Prob.  40.  A  cylindrical  vessel  2  feet  in  diameter  and  3  feet 
deep  is  three-fourths  full  of  water,  and  is  revolved  about  its 
i  vertical  axis  so  that  the  water  is  just  on  the  point  of  overflow- 
ing around  the  upper  edge.  Find  the  theoretic  velocity  of 
efflux  from  an  orifice  in  the  base  at  a  distance  of  9  inches  from 
the  axis.  Ans.  12.28  feet  per  second. 

ARTICLE  30.  THE  PATH  OF  A  JET. 

When  a  jet  of  water  issues  from  a  small  orifice  in  the  ver- 
tical side  of  a  vessel  or  reservoir,  its  di- 
rection at  first  is  horizontal,  but  the 
force  of  gravity  immediately  causes  the 
jet  to  move  in  a  curve  which  will  be 
shown  to  be  the  common  parabola. 
Let  x  be  the  abscissa  and  y  the  ordi- 
nate  of  any  point  of  the  curve,  meas- 
ured from  the  orifice  as  an  origin,  as  FIG  M 
seen  in  Fig  22.  The  effect  of  the  im- 
pulse at  the  orifice  is  to  cause  the  space  x  to  be  described 
uniformly  in  a  certain  time  £,  or,  if  v  be  the  velocity  of  flow, 
x  =  vt.  The  effect  of  the  force  of  gravity  is  to  cause  the 
space  y  to  be  described  in  accordance  with  the  laws  of  falling 
bodies  (Art.  6),  or  y  =  %gf.  Eliminating  t  from  these  two 

equations  gives 

gx          x 

=  M*-^' 

which  is  the  equation  of  a  parabola  whose  axis  is  vertical  and 
whose  vertex  is  at  the  oi'fice. 

The  horizontal  range  of  the  jet  for  any  given  ordinate  y 
is  found  from  the  equation  x*  =  ^hy.  If  the  height  of  the 
vessel  be  /,  the  horizontal  range  on  the  plane  of  the  base  is 


x  =  2  Vh(l  -  h). 


62 


THE  ORE  TIC  A  L   H  YD  KA  ULICS. 


[CHAP.  II L 


This  value  is  o  when  h  =  o  and  also  when  h  —  /,  and  it  is  a 
maximum  when  h  —  \L  Hence  the  greatest  range  is  from  an. 
orifice  at  the  mid-height  of  the  vessel. 

A  more  general  case  is  that  where  the  side  of  the  vessel  is 

inclined  to  the  vertical  at  the 
angle  8,  as  in  Fig.  23.  Here  the 
jet  at  first  issues  perpendicularly 
to  the  side,  and  under  the  action 
of  the  impulsive  force  a  particle 
of  water  would  describe  the  dis- 
tance AB  in  a  certain  time  t. 
But  in  that  same  time  the  force 
of  gravity  causes  it  to  descend  through  the  distance  BC.  Now 
let  x  be  the  horizontal  abscissa  and  y  the  vertical  ordinate  of 
the  point  C  measured  from  the  origin  A.  Then  AB  =  x  sec  0% 
and  BC  =  x  tan  8  —  y.  Hence 


FIG.  23. 


x  tan  8  —  y  = 


x  sec  8  = 


The  elimination  of  t  from  these  expressions  gives,  after  replac- 
ing V*  by  its  value  2gh, 


y  =  x  tan  8  — 


(16) 


which  is  also  the  equation  of  a  common  parabola. 

To  find  the  horizontal  range  in  the  level  of  the  orifice  make 
y  •=.  o ;   then 

.    tan0 


x  — 


sec*  6 


—  2/1  sin  20. 


This  is  o  when  8  =0°  or  0  =  90°;  it  is  a  maximum  and  equal  to 
2/1  when  8  ==  45°.  To  find  the  highest  point  of  the  jet  the 
first  derivative  of  y  with  reference  to  x  is  to  be  equated  to  zero 


ART.  30.]  THE   PATH   OF  A  JET.  63 

in  order  to  locate  the  point  where  the  tangent  to  the  curve  is 
horizontal ;  thus, 

dy  x  sec2  0 

-^-  =  tan  B -7 —  =  o, 

dx  2h 

from  which  x  =  2h  sin  0  cos  6,  and  this,  inserted  in  the  equation 
of  the  curve,  gives 

y  =  h  sina  0, 

whicrj  is  the  highest  elevation  of  the  jet  above  the  orifice.  In 
this,  if  0  =  90°,  y  =  h  ;  that  is,  if  a  jet  be  directed  vertically  up- 
ward it  will,  theoretically,  rise  to  the  height  of  the  level  of 
water  in  the  reservoir. 

As  a  numerical  example  let  a  vessel  whose  height  is  16  feet 
stand  upon  a  horizontal  plane  DE,  Fig.  23,  the  side  of  the 
vessel  being  inclined  to  the  vertical  at  the  angle  B  •=  30°.  Let 
a  jet  issue  from  a  small  orifice  at  A,  under  a  head  of  10 
feet.  The  jet  rises  to  its  maximum  height,  y  —  Jio  =  2.5  feet, 
at  the  distance  x  =  -J-  ^3  X  10  —  8.66  feet  from  A.  At  x  — 
17.32  feet  the  jet  crosses  the  horizontal  plane  through  the 
orifice.  To  find  the  point  where  it  strikes  the  plane  DE,  the 
value  of  -y  is  made  —  6  feet ;  then,  from  the  equation  of  the 
curve, 

,-       ** 

-6  =  ,l/i--, 

from  which  x  is  found  to  be  24.62  feet ;  whence,  finally, 
DE  —  24.62  —  6  tan  30°  =  21.16  feet. 

Prob.  41.  Find  all  the  circumstances  of  the  motion  of  a  jet 
which  issues  from  a  vessel  under  a  head  of  5  feet,  the  side  of 
the  vessel  being  inclined  to  the  vertical  at  an  angle  of  60°,  and 
its  depth  being  9  feet. 


64  THEORETICAL  HYDRAULICS.  [CHAP.  III. 


ARTICLE  31.  THE  ENERGY  OF  A  JET. 

Let  a  jet  or  stream  of  water  have  the  velocity  v,  and  let  W 
be  the  weight  of  water  per  second  passing  any  given  cross- 
section.  The  energy  of  this  moving  water,  or  the  work  which 
it  is  capable  of  doing,  is  the  same  as  that  stored  up  by  a  body 
falling  freely  under  the  action  of  gravity  through  a  height  h 
and  thereby  acquiring  the  velocity  v.  Thus,  if  K  be  the  energy 
or  potential  work, 


K=  Wh  =  W- 


Therefore,  for  a  constant  quantity  of  water  per  second  passing 
through  the  given  cross-section,  the  energy  of  the  jet  is  pro- 
portional to  the  square  of  its  velocity. 

The  weight  W,  however,  may  be  expressed  in  terms  of  the 
cross-section  of  the  jet  and  its  velocity.  Thus,  if  a  be  the  area 
of  the  cross-section,  and  w  the  weight  of  a  cubic  unit  of  water, 
W  is  the  weight  of  a  column  of  water  whose  length  is  v  and 
whose  cross-section  is  a,  or  W=  wav\  and  hence  (17)  may  be 
written 


(17)' 


In  general,  then,  it  may  be  stated  that  for  a  constant  cross- 
section,  the  energy  of  a  jet,  or  the  work  which  it  is  capable  of 
doing  per  second,  varies  with  the  cube  of  its  velocity. 

The  expressions  just  deduced  give  the  theoretic  energy  of 
the  jet,  that  is,  the  maximum  work  which  can  be  obtained  from 
it  ;  but  this  in  practice  can  never  be  fully  utilized.  The  amount 
of  work  which  is  realized  when  a  jet  strikes  a  moving  surface, 


ART.  31.]  THE  ENERGY  OF  A  JET.  65 

like  the  vane  of  a  water-motor,  depends  upon  a  number  of  cir- 
cumstances which  will  be  explained  in  a  later  chapter,  and  it 
is  the  constant  aim  of  inventors  so  to  arrange  the  conditions 
that  the  actual  work  may  be  as  near  to  the  theoretic  energy  as 
possible.  The  "  efficiency  "  of  an  apparatus  for  utilizing  the 
energy  of  moving  water  is  the  ratio  of  the  work  actually 
utilized  to  the  theoretic  work ;  or,  if  k  be  the  work  realized, 
the  efficiency  e  is 

e-L  (18) 

*  ~  K 

The  greatest  possible  value  of  e  is  unity,  but  this  can  never  be 
attained,  owing  to  the  imperfections  of  the  apparatus  and  the 
hurtful  resistances.  Values  greater  than  0.90  have,  however, 
been  obtained  ;  that  is,  90  per  cent  or  more  of  the  theoretic 
work  has  been  utilized  in  some  of  the  best  forms  of  hydraulic 
motors. 

For  example,  let  water  issue  from  a  pipe  2  inches  in  diam- 
eter with  a  velocity  of  10  feet  per  second.  The  cross-section 

3.1416 

in  square  feet  is  - ,  and  the  theoretic  work  in  foot-pounds 

144 

per  second  is 

K  =  0.01555  X  62.5  X  0.0218  X  io3  =  21.2, 

which  is  0.0385  horse-powers.  If  the  velocity  is  100  feet  per 
second,  however,  the  theoretic  horse-power  of  the  stream  will 
be  38.5. 

Prob.  42.  One  cubic  foot  of  water  per  second  flows  from  an 
orifice  with  a  velocity  of  32  feet  per  second.  Find  the  theo- 
retic horse-power  of  the  stream. 

Prob.  43.  A  small  turbine  wheel  using  102  cubic  feet  of 
water  per  minute  under  a  head  of  40  feet  is  found  to  give  6.15 
horse-power.  Find  the  efficiency  of  the  wheel. 

Ans.  80  per  cent. 


66  THEORETICAL  HYDRAULICS.  [CHAP.  III. 

ARTICLE  32.  THE  IMPULSE  AND  REACTION  OF  A  JET. 

When  a  stream  or  jet  is  in  motion  delivering  W  pounds  of 
water  per  second  -with  the  uniform  velocity  v,  that  motion  may 
be  regarded  as  produced  by  a  constant  impulsive  force  F,  which 
has  acted  upon  W  for  one  second  and  then  ceased.  In  this 
second  the  velocity  of  F  has  increased  from  o  to  v,  and  the 
space  fyv  has  been  described.  Consequently  the  work  F  X  %v 
has  been  imparted  to  the  water  by  the  impulse  F.  But  the 

r"J 

theoretic  energy  of  the  jet  is  W  —  ;  hence 


from  which  the  force  of  impulse  F  is 

F=W-  ........      (19) 

.g 

Let  a  be  the  area  of  the  cross-section  of  the  jet  ;  then  W=  wav> 
and 


F=wa— 

o 


Therefore  the  impulse  of  a  jet  of  constant  cross-section  varies 
as  the  square  of  its  velocity. 

The  force  F  is  a  continuous  impulsive  pressure  acting  in 
the  direction  of  the  motion.  For,  by  the  definition,  F  acts  for 
one  second  upon  the  W  pounds  of  water  which  pass  a  given 
section  ;  but  in  the  next  second  W  pounds  also  pass  the  section, 
and  the  same  is  the  case  for  each  second  following.  This  im- 
pulse will  be  exerted  as  a  pressure  upon  any  surface  placed  in 
the  path  of  the  jet. 

The  reaction  of  a  jet  upon  a  vessel  occurs  when  water  flows 
from  an  orifice.  This  reaction  must  be  equal  in  value  and 
opposite  in  direction  to  the  impulse,  as  in  all  cases  of  stress 


ART.  32.]     THE  IMPULSE  AND  REACTION  OF  A  JET.  67 

action  and  reaction  are  equal.  In  the  direction  of  the  jet  the 
impulse  produces  motion,  in  the  opposite  direction  it  produces 
a  pressure  which  tends  to  move  the  vessel.  The  force  of  reac- 
tion of  a  jet  hence  is 

i)  i? 

F  =  W—=  wa  — 
g  g 

To  compare  this  with  hydrostatic  pressure,  let  h  be  the  ve- 
locity-head due  to  v ;  then 

F  =  2wa  —  =  2wak. 

But,  from  Art.  10,  the  normal  pressure  on  a  surface  of  area  a 
under  the  hydrostatic  head  h  is  wah.  Therefore  the  dynamic 
pressure  caused  by  the  reaction  of  a  jet  issuing  from  an  orifice 
in  a  vessel  is  double  the  hydrostatic  pressure  on  the  orifice 
when  closed.  This  theoretic  conclusion  has  been  verified  by 
experiment. 

The  full  force  of  impulse  or  reaction  is  exerted  in  the  line 
of  the  action  of  the  jet,  and  its  force  in  any  other  direction  is 
the  component  of  the  force  F  in  that  direction.  Hence  in  a 
direction  which  makes  an  angle  0  with  the  line  of  motion  of 
the  jet,  the  force  which  can  be  exerted  by  the  impulse  or  reac- 
tion is  .Fcos  0.  Thus  if  water  issues  from  an  orifice  in  the  base 
of  a  vessel,  it  exerts  an  upward  reaction  F  and  a  horizontal 
reaction  o  ;  if  it  issues  in  a  direction  inclined  30°  to  the  vertical, 
its  upward  reaction  is  F  cos  30°,  and  its  horizontal  reaction  is 
Fsin  30°. 

If  a  stream  moving  with  the  velocity  vl  is  retarded  so  that 
its  velocity  becomes  v9,  its  impulse  in  the  first  instance  is 

"V  "V 

W  -,  and  in  the  second  W-.     The  difference  of  these,  or 
g  S 

p=  w1^^- 


68  THEORETICAL  HYDRAULICS.  [CHAP.  III. 

is  a  measure  of  the  dynamic  pressure  developed.  It  is  by 
virtue  of  the  pressure  due  to  change  of  velocity  that  turbine 
wheels  and  other  hydraulic  motors  transform  the  energy  of 
moving  water  into  useful  work. 

Prob.  44.  Devise  an  experiment  for  measuring  the  force  of 
reaction  of  a  jet  which  issues  from  an  orifice  in  the  base  or  side 
of  a  vessel. 


ARTICLE  33.  ABSOLUTE  AND  RELATIVE  VELOCITIES. 

Absolute  velocity  is  that  with  respect  to  the  earth,  and 
relative  velocity  that  with  respect  to  a  body  in  motion.  For 
instance,  if  water  issues  from  a  small  orifice  in  a  vessel  which  is 
in  motion  in  a  straight  line  with  the  uniform  velocity  u,  the 
theoretic  velocity  of  flow  relative  to  the  vessel  is  V=  Vzg/i, 
or  the  same  as  its  absolute  velocity  if  the  vessel  were  at  rest, 
V  _  v  for  no  accelerating  forces  exist  to 
change  the  direction  or  the  value 
of  g.  The  absolute  velocity  of 
flow,  however,  may  be  greater  or 
V  * — -^ftj-u  iess  than  V,  depending  upon  the 

FlG-  24-  value  of  u  and  its  direction.     To 

illustrate :  Fig.  24  shows  a  moving  vessel  from  which  water  is 
flowing  through  three  orifices.  At  A  the  direction  of  V  is 
horizontal,  and  as  the  vessel  is  moving  in  the  opposite  direction 
with  the  velocity  u,  the  absolute  velocity  of  the  water  as  it 

leaves  the  orifice  is 

v=  V-u. 

It  is  plain  that  if  the  orifice  were  in  the  front  of  the  vessel  and 
the  direction  of  Fwere  horizontal,  the  absolute  velocity  would 
be  v  =  V-\-u. 

Again,  at  B  is  an  orifice  from  which  the  water  issues  verti- 
cally with  respect  to  the  vessel  with  the  relative  velocity  Vt 


ART.  33-1      ABSOLUTE  AND  RELATIVE    VELOCITIES.  6g 

while  at  the  same  time  the  orifice  moves  horizontally  with  the 
velocity  u.  Forming  the  parallelogram,  the  absolute  velocity 
v  is  seen  to  be  the  resultant  of  Fand  u,  or 


Lastly,  at  C  is  shown  an  orifice  in  the  front  of  the  vessel  so 
arranged  that  the  direction  of  the  relative  velocity  V  makes  an 
angle  0  with  the  horizontal.  From  C  draw  Cu  to  represent 
the  velocity  «,  and  CV  to  represent  F,  and  complete  the  par- 
allelogram as  shown  ;  then  Cv,  the  resultant  of  u  and  F,  is  the 
absolute  velocity  with  which  the  water  leaves  the  orifice. 
From  the  triangle  Cuv, 


v  =  Vtf  -f  F2  +  2^  Fcos  0. 

In  this,  if  0  =  o,  v  becomes  u  -\-  V  as  before  shown  ;  if  0  =  90°, 
it  becomes  the  same  as  when  the  water  issues  vertically  from 
the  orifice  in  the  base;  and  if  0  =  180°,  the  value  of  v  is  that 
before  found  for  an  orifice  in  the  side  of  the  vessel. 

In  Art.  29  the  velocity  of  flow  from  an  orifice  in  a  vessel 
revolving  with  uniform  velocity  was  found  to  be 


This  is  the  velocity  relative  to  the  vessel.  If  the  orifice  be  in 
the  base,  the  direction  of  V  with  respect  to  the  vessel  is  ver- 
tical, and  as  the  orifice  is  moving  horizontally  with  the  uniform 
velocity  «,  the  absolute  velocity  of  flow  is 


v  =  Vu*  +  F2  =  \/2gh  +  2u\ 

In  the  same  way,  if  the  orifice  be  in  the  side  of  the  vessel,  and 
the  direction  of  V  be  horizontal  and  directly  away  from  the 
axis,  the  same  formula  applies,  for  the  absolute  velocity  v  is 
the  resultant  of  the  two  rectangular  components  Fand  u. 


7O  THEORETICAL   HYDRAULICS.  [CHAP.  III. 

If  a  vessel  move  with  a  motion  which  is  accelerated  or  re- 
tarded, this  affects  the  value  of  g,  and  the  reasoning  of  the  pre- 
ceding articles  does  not  give  the  correct  value  of  V.  For 
instance,  if  a  vessel  move  vertically  upward  with  an  accelera- 
tion /,  the  theoretic  relative  velocity  of  flow  from  an  orifice  in 
it  is 


and  if  u  be  its  velocity  at  any  instant,  the  absolute  velocity 
of  flow  is  u  -\-  V.  This  equation  shows  that  if  a  vessel  be 
moving  downward  with  the  acceleration  g,  that  is,  freely 
falling,  Fwill  be  zero,  which  of  course  is  to  be  expected  since 
both  water  and  vessel  are  alike  accelerated. 

Prob.  45.  If  V  be  velocity  of  flow  from  the  orifice  at  A  in 
Fig.   23,   show  that  the  velocity   of  the  jet  at  the   point  E 


s 

Prob.  46.  If  a  vessel  of  water  is  moving  horizontally  with 
an  acceleration  %g,  show  that  the  surface  of  the  water  is  a 
plane  which  is  inclined  to  the  horizontal  at  an  angle  of  about 
14  degrees. 


ART.  34.] 


THE   STANDARD    ORIFICE. 


CHAPTER  IV. 
FLOW  OF  WATER  THROUGH  ORIFICES. 

ARTICLE  34.  THE  STANDARD  ORIFICE. 

Orifices  for  the  measurement  of  water  are  usually  placed  in 
the  vertical  side  of  a  vessel  or  reservoir,  but  may  also  be  placed 
in  the  base.  In  the  former  case  it  is  understood  that  the 
upper  edge  of  the  opening  is  completely  covered  with  water; 
and  generally  the  head  of  water  on  an  orifice  is  at  least  three 
or  four  times  its  vertical  height.  The  term  standard  orifice 
is  here  used  to  signify  that  the  opening  is  so  arranged  that 
the  water  in  flowing  from  it  touches  only  a  line,  as  would 
be  the  case  in  a  plate  of  no  thickness.  To  secure  this  result 
the  inner  edge  of  the  opening  has  a  square  corner,  which  alone 
is  touched  by  the  water.  In  precise  experiments  the  orifice 
may  be  in  a  metallic  plate  whose 
thickness  is  really  small,  as  at  A  in 
Fig.  25,  but  more  commonly  it  is 
cut  in  a  board  or  plank,  care  being 
taken  that  the  inner  edge  is  a 
definite  corner.  It  is  usual  to  bevel 
the  outer  edges  of  the  orifice  as  at 
C,  so  that  the  escaping  jet  may  by 
no  possibility  touch  the  edges  ex- 
cept at  the  inner  corner.  The  term  "  orifice  in  a  thin  plate  "  is 
often  used  to  express  the  condition  that  the  water  shall  only 
touch  the  edges  of  the  opening  along  a  line.  This  arrange- 


FlG. 


25- 


72  FLO  IV  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. . 

ment  may  be  regarded  as  a  kind  of  standard  apparatus  for  the 
measurement  of  water,  for,  as  will  be  seen  later,  the  discharge 
is  modified  if  the  inner  corner  is  rounded,  and  different  de- 
grees of  rounding  give  different  discharges.  Orifices  arranged 
as  in  Fig.  25  are  accordingly  always  used  when  water  is  to  be 
measured  by  the  use  of  orifices. 

The  contraction  of  the  jet  which  is  always  observed  when 
water  issues  from  a  standard  orifice  as  described  above  is  a 
most  interesting  and  important  phenomenon.  It  is  due  to  the 
circumstance  that  the  particles  of  water  as  they  approach  the 
orifice  move  in  converging  directions,  and  that  these  directions 
continue  to  converge  for  a  short  distance  beyond  the  plane  of 
the  orifice.  It  is  this  contraction  of  the  jet  that  causes  only 
the  inner  corner  of  the  orifice  to  be  touched  by  the  escaping 
water.  The  appearance  of  such  a  jet  under  steady  flow,  issuing 
from  a  circular  orifice,  is  that  of  a  clear  crystal  bar  whose 
beauty  excites  the  admiration  of  every  observer.  The  place 
of  greatest  contraction  is  at  a  distance  from  the  plane  of  the 
orifice  of  about  one-half  its  diameter,  and  beyond  this  point 
the  jet  gradually  enlarges  in  size,  while  its  surface  becomes 
more  or  less  disturbed  owing  to  the  resistance  of  the  air  and 
other  causes. ,  In  the  case  of  square  and  rectangular  orifices 
the  contraction  of  the  jet  is  also  observed,  its  edges  being 
angular  and  its  cross-section  similar  to  that  of  the  orifice  until 
the  place  of  greatest  contraction  is  passed. 

Owing  to  this  contraction  the  discharge  from  a  standard 
orifice  is  always  less  than  the  theoretic  discharge.  It  is  the 
object  of  this  chapter  to  determine  how  the  theoretic  formulas 
are  to  be  modified  so  that  they  may  be  used  for  the  practical 
purposes  of  the  measurement  of  water.  This  is  to  be  done  by 
the  discussion  of  the  results  of  experiments.  It  will  be  sup- 
posed, unless  otherwise  stated,  that  the  size  of  the  orifice  is 
small  compared  with  the  cross-section  of  the  reservoir,  so  that 


ART.  35.]          THE   COEFFICIENT  OF  CONTRACTION.  ?$ 

the  effect  of  velocity  of  approach  may  be  neglected  (Art.  25). 

Prob.  47.  Under  a  head  of  6  feet  the  discharge  from  an 
orifice  is  3.74  gallons  per  second.  What  head  is  necessary  in 
order  that  the  discharge  may  be  one  cubic  foot  per  second  ? 

ARTICLE  35.  THE  COEFFICIENT  OF  CONTRACTION. 

The  coefficient  of  contraction  is  the  number  by  which  the 
area  of  the  orifice  is  to  be  multiplied  in  order  to  give  the  area 
of  the  least  cross-section  of  the  jet.  Thus,  if  c'  be  the  co- 
efficient of  contraction,  a  the  area  of  the  orifice,  and  a'  that  of 

the  jet, 

a'  =  c'a (20) 

The  coefficient  of   contraction  is  evidently  always  less  than 
unity. 

The  only  direct  method  of  finding  the  value  of  c'  is  to 
measure  by  callipers  the  dimensions  of  the  least  cross-section 
of  the  jet.  The  size  of  the  orifice  can  usually  be  determined 
with  accuracy,  but  no  great  precision  can  be  attained  in 
measuring  the  jet.  To  find  c'  for  a  circular  orifice  let  d  and  dr 
be  the  diameters  of  the  sections  a  and  a' ;  then 


Therefore  the  coefficient  of  contraction  is  the  square  of  the 
ratio  of  the  diameter  of  the  jet  to  that  of  the  orifice.  In  this 
way  NEWTON  found  for  c'  the  value  0.71  ;  BORDA,  0.65  ;  Bos- 
SUT,  from  0.66  to  0.67;  MICHELOTTI,  from  0.57  to  0.624  with 
a  mean  of  0.6 1.  EYTELWEIN  gave  0.64  as  a  mean  value,  and 
WEISBACH  mentions  0.63. 

As  a  mean  value  the  following  may  be  kept  in  mind  by  the 
student : 

Coefficient  of  contraction  c'  =  0.62  ; 


74  FLOW  OF   WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

or,  in  other  words,*  the  minimum  cross-section  of  the  jet  is  62 
per  cent  of  that  of  the  orifice.  This  value,  however,  undoubt- 
edly varies  for  different  forms  of  orifices  and  for  the  same 
orifice  under  different  heads,  but  little  is  known  regarding  the 
extent  of  these  variations  or  the  laws  that  govern  them.  Prob- 
ably c'  is  slightly  smaller  for  circles  than  for  squares,  and 
smaller  for  squares  than  for  rectangles,  particularly  if  the  rect- 
angle be  long  compared  with  its  width.  Probably  also  c'  is 
larger  for  low  heads  than  for  high  heads. 

Prob.  48.  The  diameter  of  a  circular  orifice  is  1.995  inches. 
Three  measurements  of  the  diameter  of  the  least  cross-section 
of  the  jet  give  the  values  1.55,  1.56,  and  1.59  inches.  Find  the 
coefficient  of  contraction. 

ARTICLE  36.  THE  COEFFICIENT  OF  VELOCITY. 

The  coefficient  of  velocity  is  the  number  by  which  the  theo- 
retic velocity  of  flow  from  the  orifice  is  to  be  multiplied  in 
order  to  give  the  actual  velocity  at  the  least  cross-section  of 
the  jet.  Thus,  if  cl  be  the  coefficient  of  velocity,  V  the  theo- 
retic velocity  due  to  the  head  on  the  centre  of  the  orifice,  and 
v  the  actual  velocity  at  the  contracted  section, 

V  =  C,V=C,<ftih (21) 

The  coefficient  of  velocity  must  be  less  than  unity,  since  the 
force  of  gravity  cannot  generate  a  greater  velocity  than  that 
due  to  the  head. 

The  velocity  of  flow  at  the  contracted  section  of  the  jet 
cannot  be  directly  measured.  To  obtain  the  value  of  the  co- 
efficient of  velocity,  indirect  observations  have  been  taken  on 
the  path  of  the  jet.  Referring  to  Art.  30,  it  will  be  seen  that 
when  a  jet  flows  from  an  orifice  in  the  vertical  side  of  a  vessel 
it  takes  a  path  whose  equation  is 

v-g- 

y~  2V" 


ART.  36.]  THE   COEFFICIENT  OF   VELOCITY.  75 

in  which  x  and  y  are  the  co-ordinates  of  any  point  of  the  path 
measured  from  vertical  and  horizontal  axes,  and  v  is  the  ve- 
locity at  the  origin.  Now  placing  for  v  its  value  cl  *J  2gh,  and 
solving  for  ^  ,  gives 


Therefore  cl  becomes  known  by  the  measurement  of  the  two 
co-ordinates  x  and  y  and  the  head  //. 

In  conducting  this  experiment  it  would  be  well  to  have  a 
ring,  a  little  larger  than  the  jet,  supported  by  a  stiff  frame 
which  can  be  moved  until  the  jet  passes  through  the  ring. 
The  flow  of  water  can  then  be  stopped,  and  the  co-ordinates  of 
the  centre  of  the  ring  determined.  By  placing  the  ring  at 
different  points  of  the  path  different  sets  of  co-ordinates  can  be 
obtained.  The  value  of  x  should  be  measured  from  the  con- 
tracted section  rather  than  from  the  orifice,  since  v  is  the 
velocity  at  the  former  point  and  not  at  the  latter. 

By  this  method  of  the  jet  BOSSUT  in  two  experiments 
found  for  the  coefficient  of  velocity  the  values  0.974  and  0.980, 
MlCHELOTTl  in  three  experiments  obtained  0.993,  0.998,  and 
0.983,  and  WEISBACH  deduced  0.978.  Great  precision  cannot 
be  obtained  in  these  determinations,  nor  indeed  is  it  necessary 
for  the  purposes  of  hydraulic  investigation  that  cl  should  be 
accurately  known  for  standard  orifices.  As  a  mean  value  the 
following  may  be  kept  in  the  memory  : 

Coefficient  of  velocity  r,  =  0.98  ,* 

or,  the  actual  velocity  of  flow  at  the  contracted  section  is  98 
per  cent  of  the  theoretic  velocity.  The  value  of  cl  is  greater 
for  high  than  for  low  heads,  and  may  probably  often  exceed 
0.99. 

Another  method  of  finding  the  coefficient  £,  is  to  place  the 
orifice  horizontal  so  that  the  jet  will  be  directed  vertically  up- 


76  FLOW   OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

wara  as  in  Fig.  12,  Art.  20.    The  height  to  which  it  rises  is  the 
velocity  height  /z0  ,  or 


in  which  v  is  the  actual  velocity  cl  V2gk.  Substituting,  this 
value  of  v  gives 

k.  =  c?h, 

from  which,  when  7z0  is  measured,  cl  is  computed.  For  ex- 
ample, under  a  head  of  23  feet  a  stream  was  found  to  rise  to  a 
height  of  22  feet  ;  then 


This  method,  like  the  preceding,  fails  to  give  good  results  for 
high  velocities  owing  to  the  resistance  of  the  air,  and  moreover 
it  is  impossible  to  measure  with  precision  the  height  7z0. 

Prob.  49.  MlCHELOTTl  found  the  range  of  a  jet  to  be  6.25 
meters  on  a  horizontal  plane  1.41  meters  below  the  vertical 
orifice,  which  was  under  a  head  of  7.19  meters.  Compute  the 
coefficient  of  velocity. 

ARTICLE  37.  THE  COEFFICIENT  OF  DISCHARGE. 

The  coefficient  of  discharge  is  the  number  by  which  the 
theoretic  discharge  is  to  be  multiplied  in  order  to  obtain  the 
actual  discharge.  Thus,  if  c  be  the  coefficient  of  discharge, 
Q  the  theoretical  and  q  the  actual  discharge  per  second, 

q  =  cQ  .........    (22) 

Evidently  c  is  a  number  less  than  unity. 

The  coefficient  of  discharge  can  be  accurately  found  by 
allowing  the  flow  from  an  orifice  to  fall  into  a  vessel  whose 
cubic  contents  are  known  with  precision.  The  quantity  q  is 


ART.  37-]  THE   COEFFICIENT  OF  DISCHARGE.  J? 

thus  determined,  while  Q  is  computed  from  the  formulas  of  the 
last  chapter.     Then 


For  example,  a  circular  orifice  of  o.i  feet  diameter  was  kept 
under  a  constant  head  of  4.677  feet;  during  a  time  of  5  minutes 
32-^  seconds  the  jet  flowed  into  a  measuring  vessel  which  was 
found  to  contain  27.28  cubic  feet.  Here  the  actual  discharge 
per  second  was 

?7  °  S 

q  =    ''"     =  0.08212  cubic  feet. 
332.2 

The  theoretic  discharge,  from  formula  (8),  is 


Q  —  n  x  0.05"  X  8.02  1/4.677  =0.1361  cubic  feet. 
Then,  for  the  coefficient  of  discharge, 

0.08212  , 

c  =  —  =  0.604. 

0.1361 

In  this  manner  thousands  of  experiments  have  been  made 
upon  different  forms  of  orifices  under  different  heads,  for  ac- 
curate knowledge  regarding  this  coefficient  is  of  great  impor- 
tance in  practical  hydraulic  work. 

The  following  articles  contain  values  of  the  coefficient  of 
discharge  for  different  kinds  of  orifices,  and  it  will  be  seen 
that  in  general  c  is  greater  for  low  heads  than  for  high  heads, 
greater  for  rectangles  than  for  squares,  and  greater  for  squares 
than  for  circles.  Its  value  ranges  from  0.59  to  0.63  or  higher, 
and  as  a  mean  to  be  kept  in  mind  by  the  student  there  may 
be  stated  : 

Coefficient  of  discharge  c  =  0.61  ; 


78  FLOW  OF    WATER    THROUGH  OXIFICES.      [CHAP.  IV. 

or,  the  actual  discharge  from  orifices  such  as  are  shown  in  Fig. 
25  is  61  per  cent  of  the  theoretic  discharge. 

The  coefficient  c  may  be  expressed  in  terms  of  the  coef- 
ficients c'  and  cl .  Let  a  and  a'  be  the  areas  of  the  orifice  and 
the  cross- section  of  the  contracted  jet,  and  Q  and  q  the  theo- 
retic and  actual  discharge  per  second.  Then 

q        a'  cl  V~2gh        a' 
Q         a  V2gk         a 

But  (Art.  34)  the  ratio  a'  :  a  is  the  coefficient  c' ;  therefore 

c  =  c'c, ; (23) 

or,  the  coefficient  of  discharge  is  the  product  of  the  coefficients 
of  contraction  and  velocity. 

Prob.  50.  What  is  the  discharge  in  gallons  per  minute  from 
a  circular  orifice  one  inch  in  diameter  under  a  head  of  12 
inches,  the  coefficient  of  discharge  being  0.609? 

Prob.  51.  The  diameter  of  a  contracted  circular  jet  was 
found  to  be  0.79  inches,  the  diameter  of  the  orifice  being 
one  inch.  Under  a  head  of  4  feet  the  actual  discharge  per 
minute  was  found  to  be  3.21  cubic  feet.  Find  the  coefficient 
of  velocity. 

ARTICLE  38.  CIRCULAR  VERTICAL  ORIFICES. 

Let  h  be  the  head  on  the  centre  of  a  vertical  circular  orifice 
whose  diameter  is  d.  The  theoretic  discharge  per  second  is 
found  from  formula  (10),  Art.  24,  by  placing  for  r  its  value  %dy 
and  the  actual  discharge  per  second  is 

2=^/^(1-^  J-T5 


J2i__  .£  _  etc.),     .     (24) 

4.  104  ^04   A  I 


ART.  38.] 


CIRCULAR    VERTICAL    ORIFICES. 


79 


in  which  c  is  the  coefficient  of  discharge.  In  case  //  becomes 
large  compared  with  d,  the  negative  terms  in  the  parenthesis 
may  be  neglected,  and 

q=c.  %7td*  V^P, (24') 

which  is  the  same  as  the  formula  for  horizontal  circular  orifices 
(Art.  21). 

The  following  table  of  values  of  c  is  abridged  from  the 
results  deduced  by  HAMILTON  SMITH,  Jr.,*  as  determined  by 
the  discussion  of  all  the  best  experiments.  The  table  applies 
only  to  standard  orifices. 

TABLE  VI.    COEFFICIENTS  FOR  CIRCULAR  VERTICAL  ORIFICES. 


Head 
h 
in  Feet. 

Diameter  of  Orifice  in  Feet. 

0.02 

0.04 

0.07 

O.I 

0.2 

0.6 

I.O 

0.4 

0.637 

0.624 

0.618 

0.6 

0.655 

.630 

.618 

.613 

0.601 

0.593 

0.8 

.648 

.626 

.615 

.610 

.601 

.594 

0.590 

I.O 

.644 

.623 

.612 

.608 

.600 

•595 

•591 

1-5 

•637 

.618 

.608 

.605 

.600 

•596 

•593 

2. 

.632 

.614 

.607 

.604 

•599 

•597 

•595 

2-5 

.629 

.612 

.605 

.603 

•599 

•598 

.596 

3- 

.627 

.611 

.604 

.603 

•599 

.598 

•597 

4- 

.623 

.609 

.603 

.602 

•599 

597 

•596 

6. 

.6l8 

.607 

.602 

.600 

•593 

•597 

•596 

8. 

.614 

.605 

.601 

.600 

.598 

•  596 

•596 

10. 

.611 

.603 

•599 

.598 

•597 

.596 

•595 

20. 

.601 

•599 

•597 

.596 

•596 

.596 

•594 

50. 

.596 

•595 

•594 

•594 

•594 

•594 

•593 

IOO. 

•593 

•  592 

•592 

•592 

•592 

<592 

•  592 

This  table  shows  that  the  coefficient  c  decreases  as  the  size 
of  the  orifice  increases,  and  that  for  diameters  less  than  0.2 


Hydraulics,  p.  59. 


8O  FLOW  OF    WATER    THROUGH   ORIFICES.      [CHAP.  IV, 

feet  it  decreases  as  the  head  increases.  It  may  be  presumed 
that  the  cause  of  this  variation  is  due  to  a  more  perfect  con- 
traction of  the  jet  for  large  heads  and  large  orifices  than  for 
small  heads  and  small  orifices. 

In  applying  the  above  coefficients  to  actual  problems,  the 
approximate  formula 

q  =  c.  ^nd*  V2gh 

may  be  used  except  for  the  values  found  above  the  horizontal 
lines  in  the  last  three  columns.  For  these,  if  precision  be  re- 
quired, the  accurate  expression  for  q  must  be  employed.  The 
error  committed  by  using  the  approximate  formula  for  the 
values  above  the  horizontal  lines  will  depend  upon  the  ratio  of 
d  to  h  ;  as  shown,  in  Art.  24,  this  error  will  be  about  two-tenths 
of  one  per  cent  when  h  =  2d,  and  about  eight-tenths  of  one 
per  cent  when  h  =  d. 

Prob.  52.  Find  from  the  table  the  coefficient  of  discharge 
for  a  circular  orifice  of  two  inches  diameter  under  a  head  of 
1.75  feet. 

Prob.  53.  Compute  the  probable  actual  discharge  through  a 
circular  orifice  of  f  inches  diameter  under  a  head  of  I  foot  3 
inches. 

ARTICLE  39.  SQUARE  VERTICAL  ORIFICES. 

Let  a  square  orifice  whose  side  is  d  be  placed  with  its  edges 
truly  parallel  and  perpendicular  to  a  horizontal  plane.  Let  /^ , 
/22,  and  h  be  the  heads  of  water  on  its  upper  edge,  lower  edge, 
and  centre,  respectively.  The  theoretic  discharge  per  second 
is  found  by  replacing  b  by  d  in  formula  (9)  of  Art.  22,  and  the 
actual  discharge  is 

q  =  c.ldV2g(k.}-h?) (25) 

Further,  as  shown  in  Art.  22,  if  h  be  large  compared  with  d, 
the  discharge  may  be  computed  by  the  simpler  formula 

q  —  c.  d*  V^h (25') 

In  both  formulas  c  is  the  coefficient  of  discharge  (Art.  36). 


ART.  39.] 


SQUARE    VERTICAL   ORIFICES. 


Si 


The  following  values  of  the  coefficient  c  have  been  taken 
from  a  more  extended  table  deduced  by  SMITH  by  an  ex- 
haustive discussion  of  experiments.  They  are  applicable  only 
to  cases  where  the  orifice  has  a  sharp  inner  edge  so  that  the 
contraction  of  the  jet  may  be  perfectly  formed  (Art.  33). 

TABLE  VII.    COEFFICIENTS  FOR  SQUARE  VERTICAL  ORIFICES. 


Head 
h 
in  Feet. 

Side  of  the  Square  in  Feet. 

O.O2 

0.04 

0.07 

O.I 

0.2 

0.6 

I.O 

0.4 

0.643 

0.628 

0.621 

0.6 

0.660 

.636 

.623 

.617 

O.6O5 

0.598 

0.8 

.652 

.631 

.620 

.615 

.605 

.600 

0.597 

I.O 

.648 

.628 

.618 

.613 

.605 

.601 

•599 

1-5 

.641 

.622 

.614 

.610 

.605 

.6O2 

.601 

2. 

.637 

.619 

.612 

.608 

.605 

.604 

.602 

2-5 

•634 

.617 

.610 

.607 

.605 

.604 

.602 

3- 

.632 

.616 

.609 

.607 

.605 

.604 

.603 

4- 

.628 

.614 

.608 

.606 

.605 

.603 

.602 

6. 

.623 

.612 

.607 

.605 

.604 

.603 

.602 

S. 

.619 

.610 

.606 

.605 

.604 

.603 

.602 

10. 

.616 

.608 

.605 

.604 

.603 

.602 

.601 

20. 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

50. 

.602 

.601 

.601 

.600 

.600 

-599 

599 

100. 

•599 

.598 

.598 

.598 

.598 

.598 

.598 

The  same  general  laws  of  variation  are  here  observed  as  for 
circular  orifices,  the  coefficient  decreasing  as  the  head  increases 
and  as  the  size  of  the  square  increases.  It  should  be  noticed 
that  the  coefficients  are  always  slightly  larger  than  those  for 
circles  of  the  same  diameter ;  this  is  perhaps  caused  by  the 
less  perfect  contraction  of  the  jet  due  to  the  corners  of  the 
square. 

The  horizontal  lines  drawn  in  the  last  three  columns  of  the 
table  indicate  the  limit  h  =  4^;  so  that  the  exact  formula  is  to 


82  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

be  used  for  cases  that  fall  above  these  lines.  The  error  in  the 
use  of  the  approximate  formula  when  h  =  3.5^  is  about  one 
tenth  of  one  per  cent,  which  is  probably  less  than  the  error  in 
applying  the  coefficient  to  any  given  orifice  in  practice.  For 
all  values  except  those  above  the  horizontal  lines  the  error  of 
the  approximate  formula  is  much  less  than  one-tenth  of  one 
per  cent. 

There  are  few  recorded  experiments  on  large  square  orifices. 
ELLIS  measured  the  discharge  from  a  vertical  orifice  2  feet 
square  in  an  iron  plate  which  furnishes  the  following  results  :  * 

For  h  =  2.07  feet,  c  =  0.611  ; 
For  h  =  3.05  feet,  ^  =  0.597; 
For  h  =  3.54  feet,  c  —  0.604; 

which  indicate  that  a  mean  value  of  about  0.6  for  c  is  all  that 
can  be  safely  stated  for  large  orifices. 

Prob.  54.  Find  from  the  table  the  coefficient  of  discharge 
for  a  square  whose  side  is  3  inches  when  the  head  on  its  centre 
is  1.8  feet. 

Prob.  55.  Compute  the  probable  actual  discharge  from  a 
vertical  orifice  one  foot  square  when  the  head  on  its  upper  edge 
is  one  foot.  Ans.  5.85  cubic  feet  per  second. 

ARTICLE  40.  RECTANGULAR  VERTICAL  ORIFICES. 

Rectangular  vertical  orifices  with  the  longest  edge  hori- 
zontal are  frequently  employed  for  the  measurement  of  water. 
If  b  be  the  breadth,  d  the  depth,  h^h^,  and  h  the  head  on  the 
upper  edge,  lower  edge,  and  centre,  and  c  the  coefficient  of  dis- 
charge, the  discharge  per  second  is 


(26) 


*  Transactions  American  Society  Civil  Engineers,  1876,  vol.  v.  p.  92. 


ART.  40.] 


RECTANGULAR    VERTICAL    ORIFICES. 


or  more  simply,  if  h  be  greater  than 

q  =  c  . 


(260 


The  following  values  of  the  coefficient  c  have  been  compiled 
and  computed  from  the  discussion  given  by  FANNING.*  Those 
above  the  horizontal  lines  are  to  be  used  in  the  exact  formula, 
and  those  below  in  the  approximate  formula. 

TABLE  VIII.  COEFFICIENTS  FOR  RECTANGULAR  ORIFICES 
i  FOOT  WIDE. 


Head 
h 
in  Feet. 

Depth  of  Orifice  in  Feet. 

0.125 

0.25 

0.50 

0.75 

I.O 

1-5 

2.O 

0.4 

0.634 

0.633 

0.622 

0.6 

•633 

.633 

.619 

0.614 

0.8 

.633 

.633 

.618 

.612 

0.6o8 

i. 

.632 

.632, 

.618 

.612 

.606 

0.626 

1-5 

.630 

.631 

.618 

.611 

.605 

.626 

0.628 

2. 

.629 

.630 

.617 

.611 

.605 

.624 

.630 

2-5 

.628 

.628 

.616 

.611 

.605 

.616 

.627 

3- 

.627 

.627 

.615 

.610 

.605 

.614 

.619 

4- 

.624 

.624 

.614 

.609 

.605 

.612 

.6l6 

6. 

.615 

.615 

.609 

.604 

.602 

.606 

.610 

8. 

.609 

.607 

.603 

.602 

.601 

.602 

.604 

10. 

.606 

.603 

.601 

.601 

.601 

.601 

.602 

20. 

.601 

.601 

.6oi 

.602 

This  table  shows  that  the  variation  of  c  with  the  "head  fol- 
lows the  same  law  as  for  circles  and  squares.  ..It  is  also  seen 
that  for  a  rectangle  of  constant  breadth  the  coefficient  of  dis- 
charge increases  as  its  depth  decreases,  from  which  it  is  to  be 
inferred  that  for  a  rectangle  of  constant  depth  the  coefficient 
increases  with  the  breadth,  and  this  is  confirmed  by  other  ex- 
periments. The  value  of  c  for  a  rectangular  orifice  is  seen  to 


*  Treatise  on  Water  Supply  Engineering,  p.  205. 


84  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

be  but  slightly  larger  than  for  a  square  whose  side  is  equal  to 
the  depth  of  the  rectangle.  In  selecting  a  coefficient  for  use 
with  an  orifice  whose  size  falls  outside  the  limits  of  the  table, 
it  should  be  borne  in  mind  that  large  orifices  have  a  smaller 
value  of  c  than  small  orifices. 

A  comparison  of  the  values  of  c  for  the  orifice  one  foot  square 
with  those  in  the  last  article  shows  that  the  two  sets  of  co- 
efficients disagree,  these  being  about  one  per  cent  greater  than 
those.  This  is  probably  due  to  the  less  precise  character  and 
smaller  number  of  experiments  from  which  they  were  deduced. 
Further  experimental  data  on  rectangular  orifices  are  needed. 

Prob.  56.  What  head  is  required  to  discharge  5  cubic  feet 
per  second  through  an  orifice  3  inches  deep  and  12  inches 
long? 

Prob.  57.  What  is  a  probable  coefficient  of  discharge  for  an 
orifice  3  inches  deep  and  6  inches  long ;  also  for  an  orifice  I 
inch  deep  and  6  inches  long? 

ARTICLE  41.  THE  MINER'S  INCH. 

The  miner's  inch  may  be  roughly  defined  to  be  the  quantity 
of  water  which  will  flow  from  a  vertical  standard  orifice  one 
inch  square,  when  the  head  on  the  centre  of  the  orifice  is  6J 
inches.  From  Table  VII  the  coefficient  of  discharge  is  seen  to 
be  about  0.623,  and  accordingly  the  actual  discharge  in  cubic 
feet  per  second  is 

0.623  X  8.02      /6^ 

a  = —        -  A  /  -  -  =  0.0255, 

144  y   12 

and  the  discharge  in  one  minute  is 

60  X  0.0255  =  1.53  cubic  feet. 

The  mean  value  of  one  miner's  inch  is  therefore  about  1.5  cubic 
feet  per  minute. 

The  actual  value  of  the  miner's  inch,  however,  differs  con- 


ART.  41.]  THE  MINER'S  INCH.  8$ 

siderably  in  different  localities.  BOWIE  states  that  in  different 
counties  of  California  it  ranges  from  1.20  to  1.76  cubic  feet  per 
minute.*  The  reason  for  these  variations  is  due  to  the  fact 
that  when  water  is  bought  for  mining  or  irrigating  purposes 
a  much  larger  quantity  than  one  miner's  inch  is  required,  and 
hence  larger  orifices  than  one  square  inch  are  needed.  Thus, 
at  Smartsville  a  vertical  orifice  or  module  4  inches  deep  and 
250  inches  long,  with  a  head  of  7  inches  above  the  top  edge, 
is  said  to  furnish  1000  miner's  inches.  Again,  at  Columbia 
Hill,  a  module  12  inches  deep  and  I2f  inches  wide,  with  a  head 
of  6  inches  above  the  upper  edge,  is  said  to  furnish  200  miner's 
inches.  In  Montana  the  customary  method  of  measurement 
is  through  a  vertical  rectangle,  one  inch  deep,  with  a  head  on 
the  centre  of  the  orifice  of  4  inches,  and  the  number  of  miner's 
inches  is  said  to  be  the  same  as  the  number  of  linear  inches  in 
the  rectangle ;  thus  under  the  given  head  an  orifice  one  inch 
deep  and  60  inches  long  would  furnish  60  miner's  inches.  The 
discharge  of  this  is  said  to  be  about  1.25  cubic  feet  per  minute, 
or  75  cubic  feet  per  hour. 

A  module  is  an  orifice  which  is  used  in  selling  water,  and 
which  under  a  constant  head  is  to  furnish  a  given  number  of 
miner's  inches,  or  a  given  quantity  per  second.  The  sizes  and 
proportions  of  modules  vary  greatly  in  different  localities,  but 
in  all  cases  the  important  feature  to  be  observed  is  that  the 
head  should  be  maintained  nearly  constant  in  order  that  the 
consumer  may  receive  the  amount  of  water  for  which  he  bar- 
gains and  no  more. 

The  simplest  method  of  maintaining  a  constant  head  is  by 
placing  the  module  in  a  chamber  which  is  provided  with  a  gate 
that  regulates  the  entrance  of  water  from  the  main  reservoir  or 
canal.  This  gate  is  raised  or  lowered  by  an  inspector  once  or 
twice  a  day  so  as  to  keep  the  surface  of  the  water  in  the  cham- 

*  BOWIE,  Treatise  on  Hydraulic  Mining,  p.  268. 


•86  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV 

her  at  a  given  mark.  This  plan  though  simple  is  costly,  except 
in  works  where  many  modules  are  used,  and  where  a  daily  in- 
spection is  necessary  in  any  event,  and  it  is  not  well  adapted 
to  cases  where  there  are  frequent  and  considerable  fluctuations 
in  the  surface  of  the  water  in  the  feeding  canal. 

Numerous  methods  have  been  devised  to  secure  a  constant 
head  by  automatic  appliances  ;  for  instance,  the  gate  which 
admits  water  into  the  chamber  may  be  made  to  rise  and  fall 
by  means  of  a  float  upon  the  surface ;  the  module  itself  may 
be  made  to  decrease  in  size  when  the  water  rises,  and  to  in- 
crease when  it  falls,,  by  a  gate  or  by  a  tapering  plug  which 
moves  in  and  out  and  whose  motion  is  controlled  by  a  float. 
These  self-acting  contrivances,  however,  are  liable  to  get  out 
of  order,  and  require  to  be  inspected  more  or  less  frequently.* 

The  use  of  the  miner's  inch,  or  of  a  module,  as  a  standard 
for  selling  water,  may  be  said  to  have  a  certain  advantage  in 
simplicity,  as  it  depends  merely  upon  an  arbitrary  definition. 
It  is,  however,  greatly  to  be  desired  for  the  sake  of  uniformity 
that  water  should  be  bought  and  sold  by  the  cubic  foot.  Only 
in  this  way  can  comparisons  readily  be  made,  and  the  con- 
sumer be  sure  of  obtaining  exact  value  for  his  money. 

Prob.  58.  If  a  miner's  inch  be  1.57  cubic  feet  per  minute, 
how  many  miner's  inches  will  be  furnished  by  a  module  2 
inches  deep  and  50  inches  long  with  a  head  of  6  inches  above 
the  upper  edge  ?  f 

ARTICLE  42.  SUBMERGED  ORIFICES. 

It  is  shown  in  Art.  26  that  the  effective  head  which  causes 
the  flow  from  a  submerged  orifice  is  the  difference  in  level 
between  the  two  water  surfaces.  The  discharge  from  such  an 

*  A  cheap  and  simple  method  of  maintaining  a  nearly  constant  head  by  means 
of  an  excess  weir  is  described  by  FOOTE  in  the  Transactions  American  Society 
of  Civil  Engineers  for  March,  1887. 

\  See  BOWIE'S  Hydraulic  Mining,  page  125. 


ART.  42.] 


SUBMERGED   ORIFICES. 


orifice,  its  inner  edge  being  a  sharp  definite  corner  as  in  Fig. 
25,  has  been  found  by  experiment  to  be  somewhat  less  than 
when  the  flow  occurs  freely,  or,  in  other  words,  the  values  of 
the  coefficients  of  discharge  are  smaller  than  those  given  in 
the  preceding  articles.  The  difference,  however,  is  very  slight 
for  large  orifices  and  large  heads,  and  for  orifices  one  inch 
square  under  six  inches  head  is  about  2  per  cent. 

The  following  table  gives  values  of  the  coefficient  of  dis- 
charge for  submerged  orifices  as  determined  by  the  experi- 
ments of  HAMILTON  SMITH,  Jr.  The  height  of  the  water  on 
the  exterior  of  the  orifices  varied  from  0.57  to  0.73  feet  above 
their  centres. 

TABLE  IX.  COEFFICIENTS  FOR  SUBMERGED  ORIFICES 


Effective 
Head  in  Feet. 

Size  of  Orifice  in  Feet. 

Circle 
0.05 

Square 
0.05 

Circle 

O.I 

Square 

0.1 

Rectangle 
0.05  x  0.3 

0-5 

0.615 

0.619 

o  603 

0.6o8 

o  623 

1.0 

.610 

.614 

602 

.606 

•622 

1-5 

607 

.612 

.600 

.605 

.621 

2.0 

.605 

.610 

-599 

.604 

.  620 

2-5 

.603 

.608 

•598 

.604 

.619 

3-o 

.602 

.607 

.598 

.604 

.618 

4.0 

.601 

.606 

.598 

.604 

The  theoretic  discharge  from  a  submerged  orifice  is  the 
same  for  the  same  effective  head  whatever  be  its  distance  be- 
low the  lower  water  level.  It  is  not  likely,  however,  that  the 
same  coefficients  of  discharge  would  be  found  for  deeply  sub- 
merged orifices  as  for  those  submerged  but  slightly.  Experi- 
ments in  this  direction  from  which  to  draw  conclusions  are 
lacking. 

Prob.  59.  An  orifice  one  inch  square  in  a  gate  such  as  shown 
in  Fig.  7,  Art.  14,  is  3  feet  below  the  higher  water  level  and  2 


88  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

feet  below  the  lower  water  level.     Compute  the  discharge  in 
cubic  feet  per  minute.  Ans.  2.04  cubic  feet. 

ARTICLE  43.  SUPPRESSION  OF  THE  CONTRACTION. 

When  a  vertical  orifice  has  its  lower  edge  at  the  bottom  of 
the  reservoir,  as  shown  at  A  in  Fig.  26,  the  particles  of  water 
flowing  through  its  lower  portion  move  in 
lines  nearly  perpendicular  to  the  plane  of 
the  orifice,  or  the   contraction  of  the  jet 
does   not   form  on  the  lower   side.     This 
is  called  a  case  of  suppressed  or  incom- 
plete contraction.     The  same  thing  occurs, 
but  in  a  lesser  degree,  when  the  lower  edge 
of  the  orifice  is  near  the  bottom  as  shown 
at  B.     In  like  manner,  if  an  orifice  be  placed  so  that  one  of  its 
vertical  edges  is  at  or  near  a  side  of  the  reservoir,  as  at  C, 
the  contraction  of  the  jet  is  suppressed  upon  one  side,  and  if 
it  be  placed  at  the  lower  corner  of  the  reservoir,  suppression 
occurs  both  upon  one  side  and  the  lower  part  of  the  jet. 

The  effect  of  suppressing  the  contraction  is,  of  course,  to 
increase  the  cross-section  of  the  jet  at  the  place  where  full  con- 
traction would  otherwise  occur,  and  it  is  found  by  experiment 
that  the  discharge  is  likewise  increased.  Experiments  also 
show  that  more  or  less  suppression  of  the  contraction  will 
occur  unless  each  edge  of  the  orifice  is  at  a  distance  at  least 
equal  to  three  times  its  least  diameter  from  the  sides  or  bottom 
of  the  reservoir. 

The  experiments  of  LESBROS  and  BlDONE  furnish  the 
means  of  estimating  the  increased  discharge  caused  by  sup- 
pression of  the  contraction.  They  indicate  that  for  square 
orifices  with  contraction  suppressed  on  one  side  the  coefficient 
of  discharge  is  increased  about  3.5  per  cent,  and  with  contrac- 
tion suppressed  on  two  sides  about  7.5  per  cent.  For  a  rect- 


ART.  44.] 


ORIFICES    WITH  ROUNDED  EDGES. 


89 


angular  orifice  with  the  contraction  suppressed  on  the  bottom 
edge  the  percentages  are  larger,  being  about  6  or  7  per  cent 
when  the  length  of  the  rectangle  is  four  times  its  height,  and 
from  8  to  12  per  cent  when  the  length  is  twenty  times  the 
height.  The  percentage  of  increase,  moreover,  varies  with 
the  head,  the  lowest  heads  giving  the  lowest  percentages. 

It  is  apparent  that  suppression  of  the  contraction  should 
be  avoided  if  accurate  results  are  desired.  The  experiments 
from  which  the  above  conclusions  are  deduced  were  made  upon 
small  orifices  with  heads  less  than  6  feet,  and  it  is  not  known 
how  they  will  apply  to  large  orifices  under  high  heads. 

Prob.-6o.  Compute  the  probable  discharge  from  a  vertical 
orifice  one  foot  square  when  the  head  on  its  upper  edge  is  one 
foot,  the  contraction  being  suppressed  on  the  lower  edge. 

ARTICLE  44.  ORIFICES  WITH  ROUNDED  EDGES. 

If  the  inner  edge  of  the  orifice  be  rounded,  as  shown  in  Fig. 
27,  the  contraction  of  the  jet  is  modified,  and  the  discharge  is 
increased.  With  a  slight  degree  of 
rounding,  as  at  A,  a  partial  contrac- 
tion occurs ;  but  with  a  more  com- 
plete rounding,  as  at  C,  the  parti- 
cles of  water  issue  perpendicular  to 
the  plane  of  the  orifice  and  there  is 
no  contraction  of  the  jet.  If  a  be 
the  area  of  the  least  cross-section  of  FIG.  27. 

the  orifice,  and  a'  that  of  the  jet,  the  coefficient  of  contraction 
(Art.  34)  is 


Fora  standard  square-edged  orifice  (Fig.  25)  the  mean  value  of 
cf  is  0.62,  but  with  a  rounded  orifice  c'  may  have  any  value  be- 
tween 0.62  and  i.o,  depending  upon  the  degree  of  rounding. 


9O  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

The  coefficient  of  discharge  for  square-edged  orifices  has  a 
mean  value  of  about  0.61  ;  this  is  increased  with  rounded  edges 
and  may  have  any  value  between  0.6 1  and  i.o,  although  it  is 
not  probable  that  values  greater  than  0.95  can  be  obtained 
except  by  the  most  careful  adjustment  of  the  rounded  edges  to 
the  exact  curve  of  a  completely  contracted  jet. 

A  rounded  interior  edge  in  an  orifice  is  therefore  always  a 
source  of  error  when  the  object  of  the  orifice  is  the  measure- 
ment of  the  discharge.  If  a  contract  provides  that  water  shall 
be  gauged  by  standard  orifices,  care  should  always  be  taken 
that  the  interior  edges  do  not  become  rounded  either  by  acci- 
dent or  by  design. 

Prob.  61.  If  an  orifice  with  rounded  edges  has  a  coefficient  of 
contraction  01*0.85  and  a  coefficient  of  discharge  of  0.75,  find 
the  coefficient  of  velocity. 

ARTICLE  45.  THE  MEASUREMENT  OF  WATER  BY  ORIFICES. 

In  order  that  water  may  be  accurately  measured  by  the  use 
of  orifices  many  precautions  must  be  taken,  some  of  which 
have  already  been  noted,  but  may  here  be  briefly  recapitulated. 
The  area  of  the  orifice  should  be  small  compared  with  the  size 
of  the  reservoir  in  order  that  velocity  of  approach  may  not 
affect  the  flow  (Art.  25).  The  inner  edge  of  the  orifice  must 
have  a  definite  right-angled  corner,  and  its  dimensions  are  to 
be  accurately  determined.  If  the  orifice  be  in  wood,  care  should 
be  taken  that  the  inner  surface  be  smooth,  and  that  it  be  kept 
free  from  the  slime  which  often  accompanies  the  flow  of  water 
even  when  apparently  clear.  That  no  suppression  of  the  con- 
traction may  occur,  the  edges  of  the  orifice  should  not  be  nearer 
than  three  times  its  least  dimension  to  a  side  of  the  reservoir. 

Orifices  under  very  low  heads  should  be  avoided,  because 
slight  variations  in  the  head  produce  relatively  large  errors, 
and  also  because  the  coefficients  of  discharge  vary  more  rapidly 


ART.  45.]          MEASUREMENT  OF    WATER  BY  ORIFICES.  9 1 

and  are  probably  not  so  well  determined  as  for  cases  where  the 
head  is  greater  than  four  times  the  depth.  For  similar  reasons 
very  small  orifices  are  not  desirable.  If  the  head  be  very  low 
on  an  orifice,  vortices  will  form  which  render  any  estimation  of 
the  discharge  unreliable. 

The  measurement  of  the  head,  if  required  with  precision, 
must  be  made  with  the  hook  gauge  which  is  described  in  Art. 
50.  For  heads  greater  than  two  or  three  feet  the  readings  of 
an  ordinary  glass  gauge  placed  upon  the  outside  of  the  reser- 
voir will  usually  prove  sufficient,  as  this  can  be  read  to  hun- 
dredths  of  a  foot  with  accuracy.  An  error  of  o.oi  feet  when  the 
head  is  3.00  feet  produces  an  error  in  the  computed  discharge 
of  less  than  two-tenths  of  one  per  cent ;  for,  the  discharges  be- 
ing proportional  to  the  square  roots  of  the  heads,  |/3x>7  divided 
by  </3X)o  ecluals  i-OOi/.  For  the  rude  measurements  in  con- 
nection with  the  miner's  inch  a  common  foot-rule  will  probably 
suffice. 

The  effect  of  temperature  upon  the  discharge  remains  to  be 
noticed  ;  this  is  only  appreciable  with  small  orifices  and  under 
low  heads.  UNWIN  found  that  the  discharge  was  diminished 
one  per  cent  by  a  rise  of  144  degrees  in  temperature ;  his  orifice 
was  a  circle  0.033  ^eet  m  diameter  under  heads  ranging  from 
i.o  to  1.5  feet.  SMITH  found  that  the  discharge  was  dimin- 
ished one  per  cent  by  a  rise  of  55  degrees  in  temperature  ;  his 
orifice  was  a  circle  0.02  feet  in  diameter,  under  heads  ranging 
from  0.56  to  3.2  feet.  This  is  a  further  reason  why  small  ori- 
fices and  low  heads  are  not  desirable  in  precise  measurements 
of  discharge. 

The  coefficients  given  in  the  preceding  tables  may  be  sup- 
posed liable  to  a  probable  error  of  two  or  three  units  in  the 
third  decimal  place;  thus  a  coefficient  0.615  should  really  be 
written  0.615  ±  0.003  '•>  tnat  is>  tne  actual  value  is  as  likely  to 
be  between  0.612  and  0.618  as  to  be  outside  of  those  limits. 


92  FLOW  OF    WATER    THROUGH   ORIFICES.      [CHAP.  IV. 

The  probable  error  in  computed  discharges  due  to  the  coeffi- 
cient is  hence  about  one-half  of  one  per  cent.  To  this  are 
added  the  errors  due  to  inaccuracy  of  observation,  so  that  it  is 
thought  that  the  probable  error  of  careful  work  with  standard 
circular  orifices  is  at  least  one  per  cent.  The  computed  dis- 
charges are  hence  liable  to  error  in  the  third  significant  figure,  so 
that  it  is  useless  to  carry  numerical  results  beyond  four  figures 
when  based  upon  tabular  coefficients.  As  a  precise  method  of 
measuring  small  quantities  of  water,  standard  orifices  take  a 
high  rank  when  the  observations  are  conducted  with  care. 
With  rectangular  orifices  the  probable  error  is  liable  to  be  two 
per  cent  or  more. 

Prob.  62.  What  error  is  produced  in  the  computed  discharge 
if  the  head  be  read  1.38  feet  when  it  should  have  been  1.385 
feet? 

ARTICLE  46.  THE  ENERGY  OF  THE  DISCHARGE. 

A  jet  of  water  flowing  from  an  orifice  possesses  by  virtue 
of  its  velocity  a  certain  energy  or  potential  work,  which  is  al- 
ways less  than  the  theoretic  energy  due  to  the  head  (Art.  31). 
Let  h  be  the  head  and  W  the  weight  of  water  discharged  per 
second,  then  the  theoretic  energy  per  second  is 

K=  Wh. 

Let  v  be  the  actual  velocity  of  the  water  at  the  contracted  sec- 
tion of  the  jet ;  then  the  actual  energy  per  second  of  the  water 
as  it  passes  that  section  is 


But  -  -  is  less  than  h  because  v  is  less  than  the  theoretic  ve- 

*g 
locity ;  or,  if  cl  be  the  coefficient  of  velocity  (Art.  36), 


ART.  46.]  THE  ENERGY  OF   THE  DISCHARGE.  93 

V* 
whence  —  =  c*h  ; 

o  *  % 

and  hence  the  effective  energy  is 

k  =  c?Wh  ........     (270 

The  efficiency  of  the  jet  accordingly  is 

e  =  ~K  =  C*  ' 
which  is  always  less  than  unity. 

For  the  standard  orifice  with  square  inner  edges  a  mean 
value  of  cl  is  0.98.  The  mean  effective  energy  of  the  jet  at  the 
contracted  section  is  hence 


that  is,  the  effective  energy  is  96  per  cent  of  the  theoretic.  For 
high  heads  cl  is  greater  than  0.98,  and  the  efficiency  becomes 
greater  than  96  per  cent.  It  is  not  possible  in  practice  to  take 
advantage  of  this  high  efficiency,  on  account  of  the  difficulty  of 
placing  the  vanes  of  a  hydraulic  motor  so  near  the  orifice,  and 
accordingly  standard  orifices  are  never  used  when  the  work  of 
the  discharge  is  to  be  utilized. 

The  loss  of  energy,  or  potential  work,  is  hence  about  4  per 
cent  with  the  standard  orifice.  This  is  caused  by  the  influence 
of  the  edges  of  the  orifice  which  retard  the  velocity  of  the 
outer  filaments  of  the  jet.  That  these  outer  filaments  move 
slower  than  the  central  ones  may  be  seen  by  placing  fine  sand 
or  sawdust  in  the  water  and  observing  that  the  greater  part 
passes  out  of  the  orifice  in  the  interior  of  the  jet. 

Prob.  63.  Prove  that  the  energy  due  to  the  velocity  of  the 
jet  in  the  plane  of  the  inner  edge  of  the  standard  orifice  is 
about  37  per  cent  of  the  theoretic  energy.  How  is  the  remain- 
ing 63  per  cent  accounted  for? 


94  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

ARTICLE  47.  DISCHARGE  UNDER  A  DROPPING  HEAD. 

If  a  vessel  or  reservoir  receives  no  inflow  of  water  while  an 
orifice  is  open,  the  head  drops  and  the  discharge  decreases  in 
each  successive  second.  Let  H  be  the  head  on  the  orifice  at  a 
certain  instant,  and  h  the  head  t  seconds  later  ;  let  A  be  the 
area  of  the  uniform  horizontal  cross-section  of  the  vessel,  and  a 
the  area  of  the  orifice.  Then,  as  demonstrated  in  Art.  28,  the 
time  t  is 


=  —(VH-  Vh). 


This  is  the  theoretic  time  ;  to  determine  the  actual  time  the 
coefficient  of  discharge  must  be  introduced.  Referring  to  the 
demonstration,  it  is  seen  that  a  ^2gy  .  dt  is  the  theoretic  dis- 
charge in  the  time  dt',  hence  the  actual  discharge  is  c  .  a  V2gy  $t> 
and  accordingly  the  above  equation  is  to  be  thus  modified  : 

2  A 


ca 


(28) 


which  is  the  practical  formula  for  the  time  in  Avhich  the  water 
level  drops  from  H  to  //.  In  using  this  formula  c  may  be  taken 
from  the  tables  in  the  preceding  articles,  an  average  value  being 
selected  corresponding  to  the  average  head. 

Experiments  have  been  made  to  determine  the  value  of  c 
by  the  help  of  this  formula  ;  the  liquid  being  allowed  to  flow, 
At  a,  H,  h,  and  t  being  observed,  whence  c  is  computed.  In 
this  way  c  for  mercury  has  been  found  to  be  about  0.62.*  Only 
approximate  mean  values  can  be  found  in  this  manner,  since  c 
varies  with  the  head,  particularly  for  small  orifices  (Art.  38). 
For  a  large  orifice  the  time  of  descent  is  usually  so  small  that 
it  cannot  be  noted  with  precision,  and  the  friction  of  the  liquid 

*DOWNING'S  Elements  of  Practical  Hydraulics  (London,  1875),  p.  187. 


ART.  47.]       DISCHARGE    UNDER  A    DROPPING  HEAD.  9$ 

on  the  sides  of  the  vessel  may  also  introduce  an  element  of  un- 
certainty. This  experiment  has  therefore  little  value  except 
as  illustrating  and  confirming  the  truth  of  the  theoretic  formulas. 

The  discharge  in  one  second  when  the  head  is  H  at  the 
beginning  of  the  second  is  found  as  follows  :  The  above  equa- 
tion may  be  written  in  the  form 


. 

By  squaring  both  members,  transposing  and  multiplying  by  A, 
this  becomes 


But  the  first  term  of  this  equation  is  the  quantity  discharged 
in  t  seconds  ;  therefore  the  discharge  Q  for  t  seconds  may  be 
written 


and  the  discharge  in  one  second  is 

-*£-).  .  (29) 


If  A  =  oo,  this  becomes  ca  V2gH,  which  should  be  the  case, 
for  then  H  would  remain  constant.    The  head  at  the  end  of  one 

second  is  h  =  H j,  and  at  the  end  of  t  seconds  is h  =  H -7. 

A  JT1 

For  example,  let  an  orifice  one  foot  square  in  a  reservoir  of 
10  square  feet  section  be  under  a  head  of  9  feet.  The  orifice 
having  a  sharp  inner  corner,  the  coefficient  of  discharge  from 
Table  VII  is  0.602.  Then  the  discharge  in  one  second  is  13.9 
cubic  feet,  and  the  head  drops  to  7.61  feet.  The  discharge  in 


g6  FLOW  OF    WATER    THROUGH  ORIFICES.      [CHAP.  IV. 

the  second  second  is  12.7  cubic  feet,  and  the  head  drops  to  6.34 
feet,  and  so  on.  The  time  required  to  discharge  a  given 
quantity  may  be  found  from  the  formula  for  Q  by  solving  for  /, 
or  preferably  from  the  first  formula,  h  being  computed  from  the 
given  data. 

It  is  shown  in  Art.  25  that  if  the  head  be  maintained  con- 
stant, the  theoretic  velocity  of  flow  is 


Hence  the  actual  discharge  may  be  written 


This  furnishes  another  method  of  computing  the  discharge 
under  a  dropping  or  rising  head,  when  the  heads  are  determined 
by  observations  at  uniform  intervals,  as  is  usually  the  case  in 
practice.  The  discharge  per  second  may  be  computed  from 
this  formula,  or,  if  the  orifice  be  small,  from, 

q  =  c .  a  V2gh  , 

taking  h  as  constant  during  one  second.  By  computing  suc- 
cessive values  of  q  corresponding  to  successive  observed  values 
of  h,  the  variation  in  the  discharge  is  thus  found.  It  is  not 
advisable,  however,  to  allow  the  head  to  drop  or  rise  rapidly 
in  hydraulic  measurements.  When  such  cases  occur  h  should 
be  observed  at  least  every  half-minute ;  the  values  of  q  com- 
puted from  these  readings  should  be  plotted  on  cross-section 
paper,  and  the  curve  drawn  through  the  points  then  shows  the 
law  of  variation,  and  intermediate  values  can  be  obtained  with- 
out the  necessity  of  computation. 


ART.  43.]     EMPTYING  AND  FILLING  A    CANAL  LOCK.  97 

Prob.  64.  Find  the  time  required  to  discharge  480  gallons 
from  an  orifice  2  inches  in  diameter  at  8  feet  below  the  water 
level  in  a  tank  which  is  4  X  4  feet  in  cross-section. 


ARTICLE  48.  EMPTYING  AND  FILLING  A  CANAL  LOCK. 

A  canal  lock  is  emptied  by  opening  one  or  more  orifices  in 
the  lower  gates.  Let  a  be  their  area,  and  H  the  head  of  water 
on  them  when  the  lock  is  full  ;  let  A  be  the  area  of  the  hori- 
zontal cross-section  of  the  lock.  Then  in  the  formula  of  the 
last  article,  h  =  o,  and  the  time  of  emptying  the  lock  is 


(30) 


If  the  discharge  be  free  into  the  air,  His  the  distance  from  the 
centre  of  the  orifice  to  the  level  of  the  water  in  the  lock  when 
filled  ;  but  if,  as  is  usually  the  case,  the  orifices  be  below  the 
level  of  the  water  in  the  tail  bay,  H  is  the  difference  in  height 
between  the  two  water  levels.  The  tail  bay  is  regarded  as  so 
large  compared  with  the  lock  that  its  water  level  remains  con- 
stant. 

For  example,  let  it  be  required  to  find  the  time  of  empty- 
ing a  canal  lock  80  feet  long  and  20  feet  wide  through  two 
orifices,  each  of  4  square  feet  area,  the  head  upon  which  is  16 
feet  when  the  lock  is  filled.  Using  for  c  the  value  0.6  for  orifices 
with  square  inner  edges,  the  formula  gives 

2  X  80  X  20  X  4 
'  =    0.6  X  8  X  8.02    =  333  seconds  =  5*  mmutes' 

If,  however,  the  circumstances  be  such  that  c  is  0.8,  the  time  is 
about  250  seconds,  or  4J-  minutes.  It  is  therefore  seen  that  it 
is  important  to  arrange  the  orifices  of  discharge  in  canal  locks 
with  rounded  inner  edges  so  that  c  may  be  as  near  unity  as 


98 


FLOW  OF    WATER    THROUGH   OXIFICES.      [CHAP.  IV. 


possible,  in  order  both  to  make  the  orifices  with  their  gates  as 
small  as  practicable,  and  to  diminish  the  time  of  emptying  the 
lock. 

The  filling  of  the  lock  is  the  reverse  operation.  Here  the 
water  in  the  head  bay  remains  at  a  constant  level,  and  the  dis- 
charge through  the  orifices  in  the  upper  gates  occurs  at  first 
quickly,  diminishing  with  the  rising  head  in  the  lock.  Let  // 
be  the  effective  head  on  the  orifices  when  the  lock  is  empty, 


Head  Boi/_U~  ~_r~  ~  ~  ~ 


FIG.  28. 

and  y  the  effective  head  at  any  time  t  after  the  beginning  of 
the  discharge  into  the  lock.  The  area  of  the  section  of  the 
lock  being  A,  the  quantity  Ady  is  discharged  in  the  time  6t> 
and  this  is  equal  to  ca  \/2gy  dt,  if  a  be  the  area  of  the  orifices 
and  c  the  coefficient  of  discharge.  Hence 

Ady 

ca  V2gy ' 

and  by  integration  between  the  limits  o  and  H, 


t  — 


2A  V~H 

ca  \/~2g  ' 


which  is  the  same  as  the  formula  for  the  time  of  emptying  the 
lock.     The  times  of  filling  and  emptying  a  lock  are  therefore 


ART.  48.]     EMPTYING  AND  FILLING  A    CANAL  LOCK.  99 

equal  if  the  orifices  for  inflow  and  outflow  are  of  the  same  dimen- 
sions and  under  the  same  heads.  Usually  the  upper  orifice  is 
under  a  less  head  than  the  lower,  and  hence  its  area  must  be 
larger  if  the  time  of  filling  is  required  to  be  the  same  as  that  of 
emptying.  The  area  a  for  any  case  is  found  by  the  equation 

2AVH 

a  — -=, 

CtV2g 

in  which  A,  H,  and  t  are  given,  and  c  is  determined  from  the 
evidence  presented  in  the  preceding  pages. 

Prob.  65.  A  lock  has  a  horizontal  cross-section  of  1800  square 
feet,  and  the  lift  H  is  12  feet.  Find  the  size  of  the  orifices  for 
emptying  it  in  3  minutes  when  the  coefficient  of  discharge  is  0.7. 

Ans.  a  =  12.3  square  feet. 


100 


FLO  W  OF    WA  TER   0  VER    WEIRS. 


[CHAP.  V. 


CHAPTER   V. 
FLOW   OF   WATER   OVER   WEIRS. 

ARTICLE  49.  DESCRIPTION  OF  A  WEIR. 

A  weir  is  a  notch  in  the  top  of  the  vertical  side  of  a  vessel 
or  reservoir  through  which  water  flows.  The  notch  is  generally 
rectangular,  and  the  word  weir  will  be  used  to  designate  a  rect- 
angular notch  unless  otherwise  specified,  the  lower  edge  of  the 
rectangle  being  truly  horizontal,  and  its  sides  vertical.  The 
lower  edge  of  the  rectangle  is  called  the  "  crest"  of  the  weir. 


•   FIG.  29. 

In  Fig.  29  are  shown  the  outlines  of  two  kinds  of  weirs,  A  be- 
ing the  more  usual  form  where  the  vertical  edges  of  the  notch 
are  sufficiently  removed  from  the  sides  of  the  reservoir  or  feed- 
ing canal,  so  that  the  sides  of  the  stream  may  be  fully  con- 
tracted ;  this  is  called  a  weir  with  end  contractions.  In  the  form 
at  B,  the  edges  of  the  notch  are  coincident  with  the  sides  of 
the  feeding  canal,  so  that  the  filaments  of  water  along  the  sides 
pass  over  without  being  deflected  from  the  vertical  planes  in 
which  they  move ;  this  is  called  a  weir  without  end  contrac- 
tions, or  with  end  contractions  suppressed. 

It  is  necessary  in  order  to  make  accurate  measurements  of 
discharge  by  a  weir  that  the  same  precaution  should  be  taken 


ART.  49.] 


DESCRIPTION   OF  A    WEIR. 


101 


as  for  orifices  (Art.  34),  namely,  that  the  inner  edge  of  the 
notch  shall  be  a  definite  angular  corner  so  that  the  water 
in  flowing  out  may  touch  the  crest  only  in  a  line,  thus  insur- 
ing complete  contraction.  In  precise  observations  a  thin 
metal  plate  will  be  used  for  a  crest  as 
seen  in  Fig.  30,  while  in  common  work 
it  may  be  sufficient  to  have  the  crest 
formed  by  a  plank  of  smooth  hard 
wood  with  its  inner  corner  cut  to  a 
sharp  right  angle  and  its  outer  edge  FIG.  3o. 

bevelled.  The  vertical  edges  of  the  weir  shbvjld  'b^made  in, 
the  same  manner  for  weirs  with  end  contractions/  while  foY 
those  without  end  contractions  the  sides  of  >  tha  feeding  t^^L 
should  be  smooth  and  be  prolonged  a  slight  distance  beyond 
the  crest.  It  is  also  necessary  to  observe  the  same  precautions 
as  for  orifices  to  prevent  the  suppression  of  the  contraction 
(Art.  43),  namely,  that  the  distance  from  the  crest  of  the  weir  to 
the  bottom  of  the  feeding  canal,  or  reservoir,  should  be  greater 
than  three  times  the  head  of  water  on  the  crest.  For  a  weir 
with  end  contractions  a  similar  distance  should  exist  between 
the  vertical  edges  of  the  weir  and  the  sides  of  .the  feeding  canal. 

The  head  of  water  H  upon  the  crest  of  a  weir  is  usually 
much  less  than  the  breadth  of  the  crest,  b.  The  value  of  H 
should  not  be  less  than  o.i  foot,  and  it  rarely  exceeds  1.5  feet. 
The  least  value  of  b  in  practice  is  about  0.5  feet,  and  it  does 
not  often  exceed  20  feet.  Weirs  are  extensively  used  for 
measuring  the  discharge  of  streams,  and  for  determining  the 
quantity  of  water  supplied  to  hydraulic  motors ;  the  practical 
importance  of  the  subject  is  so  great  that  numerous  experi- 
ments have  been  made  to  ascertain  the  laws  of  flow,  and  the 
coefficients  of  discharge. 

Prob.  66.  If  a  feeding  canal  three  feet  wide  discharges  12 
cubic  feet  per  second  when  the  water  is  2  feet  deep,  what  is 
the  mean  velocity  of  flow  ? 


102 


FLOW  OF    WATER   OVER    WEIRS. 


[CHAP.  V. 


FIG. 


ARTICLE  50.  THE  HOOK  GAUGE. 

As  the  head  on  the  crest  of  a  weir  is  low  it 
must  be  determined  with  precision  in  order  to 
avoid  error  in  the  computed  discharge  (Art.  45). 
The  hook  gauge,  invented  by  BOYDEN  about 
1840,  consists  of  a  rod  sliding  vertically  in  fixed 
supports,  the  amount  of  vertical  motion  being 
determined  by  the  readings  of  a  vernier.  The 
.yernier  can  be  set  to  read  o.ooo  when  the  sharp 
point  of  the  hook  is  on  the  same  level  as  the 
crest  of  the  weir;  when  the  water  is  flowing 
over  the  crest,  the  rod  is  raised  by  the  slow- 
motion  screw  until  the  point  of  the  hook  is  at 
the  water  level.  Before  the  point  pierces  the 
surface  or  skin  of  the  water,  a  pimple  or  pro- 
tuberance is  seen  to  rise  above  it  due  to  capil- 
lary action ;  the  hook  is  then  depressed  until 
this  pimple  is  barely  perceptible,  when  the  point 
is  at  the  true  water  level.  The  head  of  water 
on  the  crest  is  then  indicated  by  the  reading 
of  the  scale  and  vernier.  The  best  hook  gauges 
are  made  to  read  to  ten-thousandths  of  a  foot, 
and  it  has  been  stated  that  an  experienced  ob- 
server can  in  a  favorable  light  detect  differences 
in  level  as  small  as  0.0002  feet.  The  surface 
of  water  at  the  hook  must  be  perfectly  quiet,, 
and  hence  a  box  without  a  bottom  or  with 
openings  to  admit  the  water  is  often  placed 
around  it.  Fig.  31  shows  the  hook  gauge  as 
arranged  by  EMERSON.* 


*  EMERSON'S  Hydrodynamics  (Springfield,  Mass.,  1881),  p.  56. 


ART.  50.]  THE  HOOK  GAUGE.  103 

A  cheaper  form  of  hook  gauge,  and  one  sufficiently  precise 
in  some  classes  of  work,  can  be  made  by  screwing  a  hook  into 
the  foot  of  a  levelling  rod.  The  back  part  of  the  rod  is  then 
held  in  a  vertical  position  by  two  clamps  on  fixed  supports, 
while  the  front  part  is  free  to  slide.  It  is  easy  to  arrange  a 
slow-motion  movement  so  that  the  point  of  the  hook  may  be  pre- 
cisely placed  at  the  water  level.  The  reading  of  the  vernier  is 
determined  when  the  point  of  the  hook  is  on  the  same  level  as 
the  crest  of  the  weir,  and  by  subtracting  from  this  the  subse- 
quent readings  the  heads  of  water  are  known.  A  New  York 
levelling  rod  reading  to  thousandths  of  a  foot  is  to  be  preferred. 

The  greatest  error  of  a  hook  gauge  is  thought  to  be  in  set- 
ting it  for  the  level  of  the  crest.  In  the  larger  forms  of  hooks 
this  may  be  done  by  taking  elevations  of  the  crest,  and  of  the 
point  of  the  hook  by  means  of  an  engineer's  level  and  a  light 
rod.  With  smaller  hooks  it  may  be  done  by  having  a  stiff 
permanent  hook  the  elevation  of  whose  point  with  respect  to 
the  crest  is  determined  by  precise  levels ;  the  water  is  then  al- 
lowed to  rise  slowly  until  it  reaches  the  point  of  this  stiff  hook, 
when  readings  of  the  vernier  of  the  lighter  hook  are  taken. 
Another  method  is  to  allow  a  small  depth  of  water  to  flow  over 
the  crest  and  to  take  readings  of  the  hook,  while  at  the  same 
time  the  depth  on  the  crest  is  measured  by  a  finely  graduated 
scale.  Still  another  way  is  to  allow  the  water  to  rise  slowly, 
and  to  set  the  hook  at  the  water  level  when  the  first  filaments 
pass  over  the  crest ;  this  method  is  not  a  very  precise  one  on 
account  of  capillary  attraction  along  the  crest.  As  the  error 
in  setting  the  hook  is  a  constant  one  which  affects  all  the  sub- 
sequent observations,  especial  care  should  be  taken  to  reduce 
it  to  a  minimum  by  taking  a  number  of  observations  from 
which  to  obtain  a  precise  mean  result. 

In  rough  gaugings  of  streams  the  precision  of  a  hook  gauge 
is  often  not  required,  and  the  heads  may  be  determined  by 


104  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

simpler  methods.  For  example,  a  post  may  be  set  with  its  top 
on  the  same  level  as  the  crest  of  the  weir,  and  the  depth  of 
water  over  the  top  of  the  post  be  measured  by  a  scale  gradu- 
ated to  tenths  and  hundredths  of  a  foot,  the  thousandths  be- 
ing either  estimated  or  omitted  entirely. 

The  head  H  on  the  crest  of  the  weir  is  in  all  cases  to  be 
measured  several  feet  up  stream  from  the  crest,  as  indicated  in 
Fig.  30.  This  is  necessary  because  of  the  curve  taken  by  the 
surface  of  the  water  in  approaching  the  weir.  The  distance  to 
which  this  curve  extends  back  from  the  weir  depends  upon 
many  circumstances  (Art.  59),  but  it  is  considered  that  perfect- 
ly level  water  will  be  found  at  2  or  3  feet  distance  back  for 
small  weirs,  and  at  6  or  8  feet  for  very  large  weirs.  It  is  de- 
sirable that  the  hook  should  be  placed  at  least  one  foot  from 
the  sides  of  the  feeding  canal,  if  possible.  As  this  is  apt  to 
render  the  position  of  the  observer  uncomfortable,  some  ex- 
perimenters have  placed  the  hook  in  a  pail  at  a  few  feet  dis- 
tance from  the  canal,  the  water  being  led  to  the  pail  by  a  pipe  : 
this  pipe  should  enter  the  feeding  canal  several  feet  above  the 
crest,  and  the  water  should  enter  it,  not  at  its  end,  but  through 
a  number  of  holes  drilled  at  intervals  along  its  circumference. 

Prob.  67.  Show  by  using  formula  (9)'  of  Art.  22  that  an 
error  of  about  one-half  of  one  per  cent  results  in  the  discharge 
if  an  error  of  o.ooi  feet  be  made  in  reading  the  head  when 
H  —  0.3  feet. 

ARTICLE  51.  FORMULAS  FOR  THE  DISCHARGE. 

The  theoretic  discharge  through  a  rectangular  notch  or 
weir  was  found  in  Art.  22  to  be 

Q  =  \V^.bH*, 

in  which  b  is  the  breadth  of  the  notch,  commonly  called  the 
length  of  the  weir,  and  H  the  depth  of  water  on  the  lower 


ART.  51.] 


FORMULAS  FOR    THE  DISCHARGE. 


10$ 


edge.  It  might  be  inferred  that  this  depth  is  that  in  the  plane 
of  the  weir  ;  but  as  the  deduction  of  the  formula  supposes 
nothing  regarding  the  fall  due  to  the  surface  curve,  and  regards 
the  velocity  at  any  point  above  the  crest  as  due  to  the  head 
upon  that  point  below  the  free  water  surface,,  it  seems  that  H 
should  be  measured  with  reference  to  that  surface,  as  is  actu- 
ally done  by  the  hook  gauge.  The  above  formula  then  gives 
the  theoretic  discharge  per  second,  provided  that  there  be  no 
velocity  at  the  point  where  H  is  measured,  which  can  only  be 
the  case  when  the  area  of  the  weir  opening  is  very  small  com- 
pared to  that  of  the  cross-section  of  the  feeding  canal.  This 
condition  would  be  fulfilled  for  a  rectangular  notch  placed  at 
the  side  of  a  large  pond. 

When  there  is  an  appreciable  velocity  of  approach  of  the 
water  at  the  point  where  H  is  measured  by  the  hook  gauge, 
the  above  formula  must  be  modified.  Let  v  be  the  mean 
velocity  in  the  feeding  canal  at  this  section  ;  this  velocity  may 
be  regarded  as  due  to  a  fall,  //,  from  the  surface  of  still  water 
at  some  distance  up  stream  from 
the  hook,  as  shown  in  Fig.  32. 
Now  the  true  head  on  the  crest  of 
the  weir  is  H-\-h,  as  this  would 
have  been  the  reading  of  the 
hook  gauge  had  it  been  placed 
where  the  water  had  no  velocity. 
discharge  is 


FlG-  32. 
Accordingly  the  theoretic 


in  which  H  is  read  by  the  hook  and  h  is  determined  from  the 
mean  velocity  v. 

The  actual  discharge  per  second  is  always  less  than  the 
theoretic  discharge,  due  to  the  contraction  of  the  stream  and 
the  resistances  of  the  edges  of  the  weir.  To  take  account  of 


106  FLO  W  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

these  a  coefficient  is  applied  to  the  theoretic  formulas  in  the 
same  manner  as  for  orifices  ;  these  coefficients  being  deter- 
mined by  experiment,  the  formulas  may  then  be  used  for 
computing  the  actual  discharge.  It  has  also  been  proposed  by 
SMITH  to  modify  the  velocity-head  hy  owing  to  the  fact  that 
the  velocity  of  approach  is  not  constant  throughout  the 
section,  but  greater  near  the  surface  than  near  the  bottom,  as 
in  streams  (Art.  107).  Accordingly  the  following  may  be 
written  as  an  expression  for  the  actual  discharge  : 


(31) 


in  which  c  is  the  coefficient  of  discharge  whose  value  is  always 
less  than  unity,  and  n  is  a  number  which  lies  between  i.o 
and  1.5.* 

The  above  formulas  are  not  in  all  respects  perfectly  satis- 
factory, and  indeed  many  others  have  been  proposed.  The 
actual  discharge  differs,  however,  so  much  from  the  theoreti- 
cal that  the  final  dependence  must  be  upon  the  coefficients 
deduced  from  experiment,  and  hence  any  fairly  reasonable 
formula  may  be  used  within  the  limits  for  which  its  coefficients 
have  been  established.  In  spite  of  the  objections  which  may 
be  raised  against  all  forms  of  formulas,  the  fact  remains  that 
the  measurement  of  water  by  weirs  is  one  of  the  most  con- 
venient methods,  and  probably  the  most  precise  method,  unless 
the  quantity  is  so  small  as  to  pass  through  a  circular  orifice 
less  than  one  foot  in  diameter.  With  proper  precautions  the 
probable  error  in  measurements  of  discharge  by  weirs  should 
be  less  than  two  or  three  per  cent. 

Prob.  68.  Find  the  velocity-head  h  when  the  mean  velocity 
of  approach  is  20  feet  per  minute. 

*  SMITH'S  Hydraulics,  p.  83. 


ART.  52.]  VELOCITY  OF  APPROACH.  IO/ 

ARTICLE  52.  VELOCITY  OF  APPROACH. 

The  velocity-head  h,  which  produces  the  mean  velocity  of 
approach  v  is  (Art.  20) 

h  =--•=.  0.01555^. 

Accordingly  to  obtain  h  the  value  of  v  must  be  determined. 
One  way  of  doing  this  is  to  observe  the  time  of  passage  of  a 
float  through  a  given  distance;  but  this  is  not  a  precise  method. 
The  usual  method  is  to  compute  v  from  an  approximate  value 
of  the  discharge,  which  is  first  computed  by  regarding  v,  and 
hence  h,  as  zero.  This  determination  is  rendered  possible  by 
the  fact  that  v  is  usually  small,  and  hence  that  h  is  quite  small 
as  compared  with  H. 

Let  B  be  the  breadth  of  the  cross-section  of  the  feeding 
canal  at  the  place  where  the  readings  of  the  hook  are  taken, 
and  let  G  be  its  depth  below  the  crest  (Fig.  32).  The  area  of 
that  cross-section  then  is 


A  = 
The  mean  velocity  in  this  section  now  is 

,-i,    III 

in  which  q'  is  found  from  the  formula 

q'  =  c%V^.bH*. 

This  value  of  q'  is  an  approximation  to  the  actual  discharge  ; 
from  it  v  is  found,  and  then  /z,  after  which  the  discharge  q  can 
be  computed.  If  thought  necessary,  h  may  be  recomputed  by 
using  q  instead  of  q'  ;  but  this  will  rarely  be  necessary. 

For  example,  the  small  weir  with  end  contractions  used  in 
the  hydraulic  laboratory  of  Lehigh  University  has  B  =  7.82 


'IOS  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

feet  and  G  =  2.5  feet.  The  length  of  the  weir  b  is  adjustable 
according  to  the  quantity  of  water  delivered  by  the  stream. 
On  April  10,  1888,  the  value  of  b  was  1.330  feet,  and  values  of 
//"ranged  from  0.429  to  0.388  feet.  It  is  required  to  find  the 
velocity  v  and  the  velocity-head  h,  when  H  •=.  0.429  feet.  Here 
the  coefficient  c  is  0.602  (Art.  53),  hence  the  approximate  dis- 
charge per  second  is 

q'  —  0.602  X  f  X  8.02  X  1-33  X  0.429!, 
or  q'  =  1.203  cubic  feet  per  second. 

The  mean  velocity  of  approach  then  is 
1.203 


from  which  the  velocity-head  h  is 

h  =  -^-—  =  0.00004  feet. 
64.32 

This  is  too  small  to  be  regarded,  since  the  hook  gauge  used 
determines  the  heads  only  to  thousandths  of  a  foot. 

The  velocity-head  h  may  be  directly  expressed  in  terms  of 

the  discharge  by  substituting  for  v  its  value  ~ ;  thus : 

A 

fg\z 

*==o.oi555(j|J .    (32) 

In  general,  this  expression  will  be  found  the  most  convenient 
one  for  computing  the  value  of  the  head  corresponding  to  the 
velocity  of  approach. 

With  a  weir  opening  of  given  size  under  a  given  head  H, 
the  velocity  of  approach  is  less  the  greater  the  area  of  the  sec- 
tion of  the  feeding  canal,  and  it  is  desirable  in  building  a  weir 
to  make  this  area  large  so  that  the  velocity  v  may  be  smalL 


ART.  53.]  WEIRS    WITH  END   CONTRACTIONS.  IOQ 

For  large  weirs,  and  particularly  for  those  without  end  con- 
tractions, v  is  sometimes  as  large  as  one  foot  per  second,  giving 
/j  =  o.oi55  feet,  and  these  should  be  regarded  as  the  highest 
values  allowable  if  precision  of  measurement  is  required. 

Prob.  69.  FTELEY  and  STEARNS'  large  suppressed  weir  had 
the  following  dimensions  :  &  =  £>  =  18.996  feet,  G=6.$$  feet, 
and  the  greatest  measured  head  was  1.6038  feet.  Taking 
c  —  0.622,  compute  the  velocity  of  approach  and  its  velocity- 
head. 

ARTICLE  53.  WEIRS  WITH  END  CONTRACTIONS. 
Let  b  be  the  breadth  of  the  notch  or  length  of  the  weir,  H 
the  head  above  the  crest  measured  by  the  hook  gauge,  and  c 
an  experimental  coefficient.     Then  if  there  be  ;io  velocity  of 
approach  the  discharge  per  second  is 


(33) 


But  if  the  mean  velocity  of  approach  at  the  section  where  the 
hook  is  placed  be  v,  let  h  be  the  head  which  would  produce 
this  velocity.  Then  the  discharge  per  second  is 


(33)' 


The  quantity  H  -\-  1.4/1  is  called  the  effective  head  on  the  crest, 
and,  as  shown  in  the  last  article,  h  is  usually  small  compared 
with  H. 

The  following  table  contains  values  of  the  coefficient  of 
discharge  c  as  deduced  by  HAMILTON  SMITH,  Jr.,*  from  a 
discussion  of  the  experiments  made  by  LESBROS,  FRANCIS, 
FTELEY  and  STEARNS,  and  others.  In  these  experiments  q  is 
determined  by  actual  measurement  in  a  vessel  of  large  size,  and 
the  other  quantities  being  observed  c  is  computed.  Values  of 
c  for  different  lengths  of  weir  and  for  different  heads  are  thus 

*  Hydraulics  (London  and  New  York,  1884),  p.  132. 


no 


FLOW   OF    WATER    OVER    WEIRS. 


[CHAP.  V. 


obtained,  which  being  plotted  enable  mean  curves  to  be  drawn, 
from  wrhich  intermediate  values  are  taken.  The  heads  in  the 
first  column  are  the  effective  heads  H-\-  1.4/1 ;  but  as  h  is  small, 
little  error  can  result  in  using  H  as  the  argument  with  which  to 
enter  the  table  in  selecting  a  coefficient. 

TABLE   X.     COEFFICIENTS   FOR   CONTRACTED   WEIRS. 


Effective 
Head 
in  Feet. 

Length  of  Weir  in  Feet. 

0.66 

i 

2 

3 

5 

10 

19 

O.I 

0.632 

0.639 

0.646 

0.652 

0.653 

0-655 

0.656 

0.15 

.619 

.625 

.634 

.638 

.640 

.641 

.642 

0.2 

.611 

.618 

.626 

.630 

.631 

.633 

•634 

0.25 
0.3 

.605 
l 
.601 

.612 
.608 

.621 
.6l6 

.624 
.619 

.626 
.621 

.628 
.624 

.629 
.625 

0.4 

•595 

.601 

.609 

.613 

.615 

.618 

.620 

0-5 

•  590 

.596 

.605 

.608 

.611 

.615 

.617 

Os6 

.587 

•593 

.6OI 

.605 

.608 

.613 

.615 

0.7 

•590 

.598 

.603 

.606 

.612 

.614 

0.8 

•595 

.6OO 

.604 

.611 

.613 

0.9 

•592 

.598 

.603 

.609 

.612 

I.O 

•590 

•595 

.601 

.608 

.611 

1.2 

.585 

.591 

•597 

.605 

.610 

1.4 

.580 

•587 

•594 

.602 

.609 

1.6 

.582 

•  591 

.600 

.607 

It  is  seen  from  the  table  that  the  coefficient  increases  with 
the  length  of  the  weir,  which  is  due  to  the  influence  of  the  end 
contractions  being  independent  of  the  length.  The  coefficient 
also  increases  as  the  head  on  the  crest  diminishes.  The  table 
also  shows  that  the  greatest  variation  in  the  coefficients  occurs 
under  small  heads,  which  are  hence  to  be  avoided  in  order  to 
secure  accurate  measurements  of  discharge. 

Interpolation  may  be  made  in  this  table  for  heads  and 
lengths  of  weirs  intermediate  between  the  values  given,  regard- 


ART.  53-]  WEIRS    WITH  END   CONTRACTIONS.  Ill 

ing  the  coefficients  as  varying  uniformly ;  but  it  will  be  better 
in  any  actual  case  to  diagram  the  coefficients  on  cross-section 
paper,  from  which  the  interpolation  can  be  made  more  easily 
and  accurately. 

As  an  example  of  the  use  of  the  formula  and  table,  let  it  be 
required  to  find  the  discharge  per  second  over  a  weir  4  feet 
long  when  the  head  H  is  0.457  feet>  there  being  no  velocity  of 
approach.  From  the  table  the  coefficient  of  discharge  is  0.614 
for  J7=o.4  and  0.6095  for  .#=0.5,  which  gives  about  0.612 
when  ff=  0.457.  Then  the  discharge  per  second  is 

q  =  0.612  X  |  X  8.02  X  4  X  O-457i  =  4.04  cubic  feet 

If  the  width  of  the  feeding  canal  be  7  feet,  and  its  depth 
below  the  crest  be  1.5  feet,  the  velocity-head  is 


h  =  0.015551 — —  i—l  =  0.00134  feet. 
V  X  l.g& 

The  effective  head  now  becomes 

H+  1.4/2  =  0.459  feet> 
and  the  discharge  per  second  is 

q  =  0.612  X  |  X  8.02  X  4  X  0.459*  —  4-°7  cubic  feet. 

It  is  to  be  observed  that  the  reliability  of  these  computed  dis- 
charges depends  upon  the  precision  of  the  observed  quanti- 
ties and  upon  the  coefficient  c;  this  is  probably  liable  to  an 
error  of  one  or  two  units  in  the  third  decimal  place,  which  is 
equivalent  to  a  probable  error  of  about  three-tenths  of  one  per 
cent.  On  the  whole,  regarding  the  inaccuracies  of  observation, 
a  probable  error  of  one  per  cent  should  at  least  be  inferred,  so 
that  the  value  q  =  4.07  cubic  feet  per  second  should  strictly  be 
written, 

q  =  4-07  ±  0.04 ; 


112  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

that  is  to  say,  the  discharge  per  second  has  4.07  cubic  feet  for 
its  most  probable  value,  and  it  is  as  likely  to  be  between  the 
values  4.03  and  4.11  as  to  be  outside  of  those  limits. 

Prob.  70.  Compute  the  discharges  per  second  through  a 
weir  whose  length  is  2.5  feet,  width  of  feeding  canal  6  feet, 
depth  below  crest  1.6  feet  when  the  heads  on  the  crest  are 
0.314,  0.315,  and  0.316  feet. 

Prob.  71.  Compute  the  coefficient  of  discharge  for  the  fol- 
lowing experiment  by  FRANCIS,  in  which  q  was  found  by  actual 
measurement  in  a  large  tank:  ^==9.997  feet,  B=  13.96  feet, 
G=4.ig  feet,  H=  1.5243  feet,  2^=64.3236  and  ^  =  61.282 
cubic  feet  per  second.  Ans.  c  =  0.602. 

ARTICLE  54.  WEIRS  WITHOUT  END  CONTRACTIONS. 

For  weirs  without  end  contractions,  or  suppressed  weirs, 
when  there  is  no  velocity  of  approach,  the  discharge  per  second 
is 

q  =  c.\&£.bH\\      .    .         .     .     (34) 

and  when  there  is  velocity  of  approach, 

(34)' 


Here  the  notation  is  the  same  as  in  the  last  article,  and  c  is  to 
be  taken  from  the  following  table,  which  gives  the  coefficients 
of  discharge  as  deduced  by  SMITH. 

It  is  seen  that  the  coefficients  for  suppressed  weirs  are 
greater  than  for  those  with  end  contractions  :  this  of  course 
should  be  the  case,  as  contractions  diminish  the  discharge. 
They  decrease  with  the  length  of  the  weir,  while  those  for 
contracted  weirs  increase  with  the  length.  Their  greatest 
variation  occurs  under  low  heads,  where  they  rapidly  increase 
as  the  head  diminishes.  It  should  be  observed  that  these 
coefficients  are  not  reliable  for  lengths  of  weirs  under  4  feet, 


ART.  54.]         WEIRS    WITHOUT  END   CONTRACTIONS.  113 

owing  to  the  few   experiments  which   have   been   made   for 
short  weirs.     Hence,  for  small  quantities  of  water,  weirs  with 


TABLE  XI.    COEFFICIENTS  FOR  SUPPRESSED  WEIRS. 


Effective 
Head 
in 
Feet. 

Length  of  Weir  in  Feet. 

19 

10 

7 

5 

4 

•       3 

a 

O.I 

0.657 

0.658 

0.658 

0.659 

0.15 

•  643 

.644 

.645 

.645 

0.647 

0.649 

0.652 

0.2 

.635 

•637 

.637 

.638 

.641 

.642 

.645 

0.25 

.630 

.632 

.633 

.634 

.636 

.638 

.641 

0-3 

.626 

.628 

.629 

.631 

.633 

.636 

.639 

0.4 

.621 

.623 

.625 

.628 

.630 

.633 

.636 

0-5 

.619 

.621 

.624 

.627 

.630 

•633 

.637 

0.6 

.618 

.620 

.623 

.627 

.630 

.634 

.638 

0.7 

.618 

.620 

.624 

.628 

.631 

.635 

.640 

0.8 

.618 

.621 

.625 

.629 

.633 

.637 

•643 

0.9 

.619 

.622 

.627 

.631 

.635 

.639 

•645 

I.O 

.619 

.624 

.628 

.633 

.637 

.641 

.648 

1.2 

.620 

.626 

.632 

.636 

.641 

.646 

1.4 

.622 

..629 

.634 

.640 

.644 

1.6 

.623 

.631 

.637 

.642 

.647 

end  contractions  should  be  built  in  preference  to  suppressed 
weirs.  For  a  weir  of  infinite  length  it  would  be  immaterial 
whether  end  contractions  existed  or  not ;  hence  for  such  a 
case  the  coefficients  lie  between  the  values  for  the  iQ-foot 
weir  in  Table  X.  and  those  for  the  iQ-foot  weir  in  the  table 
here  given. 

For  a  numerical  illustration  the  same  data  as  in  the  ex- 
ample of  the  last  article  will  be  used,  namely,  b  =  4  feet, 
G  =  1.5  feet,  and  #"—0.45 7  feet.  The  coefficient  from  the 


114  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

table  is  0.630  ;  then  for  no  velocity  of  approach  the  discharge 
per  second  is 

q  —  0.630  X  f  X  8.02  X  4  X  0.457*  =  4-16  cubic  feet. 

Here  the  width  B  would  probably  be  also  4  feet  ;  the  head 
corresponding  to  the  velocity  of  approach  then  is 

=  0.0044  feet, 


and  the  effective  head  is 

H+i\h  =  0.463  feet, 
from  which  the  discharge  per  second  is 

q  =  0.630  X  f  X  8.02  X  4  X  0.4632  =  4.24  cubic  feet. 

This  shows  that  the  velocity  of  approach  exerts  a  greater  in- 
fluence upon  the  discharge  than  in  the  case  of  a  weir  with  end 
contractions. 

Prpb.  72.  Compute  the  discharge  per  second  over  a  weir 
without  end  contractions  when  b  =  9-995  feet,  H  =  0.7955 
feet,  G  =  4.6  feet.  Ans.  q  =  23.7  cubic  feet  per  second. 

ARTICLE  55.  FRANCIS'  FORMULAS. 

The  formulas  most  extensively  used  for  computing  the 
flow  through  weirs  are  those  established  by  FRANCIS  in  1854* 
from  the  discussion  of  his  numerous  and  carefully  conducted 
experiments,  but  as  they  are  stated  without  tabular  coeffici- 
ents they  are  to  be  regarded  as  giving  only  mean  approximate 
results.  The  experiments  were  made  on  large  weirs,  most  of 
them  10  feet  long,  and  with  heads  ranging  from  0.4  to  1.6  feet, 
so  that  the  formulas  apply  particularly  to  such,  rather  than  to 
short  weirs  and  low  heads.  The  length  b  and  the  head  H  being 

*  Lowell  Hydraulic  Experiments  (4th  edition,  New  York,  1883),  p.  133. 


ART.  55.]  FRANCIS'  FORMULAS.  115 

expressed  in  feet,  the  discharge  per  second,  when  there  is  no 
velocity  of  approach,  is,  for  weirs  without  end  contractions,  or 

suppressed  weirs, 

2  =  3.33^/8; (35) 

and  for  weirs  with  end  contractions, 

9  =  3-33  (6-o.2ff)ffl (36) 

Here  it  is  regarded  that  the  effect  of  each  end  contraction  is 
to  diminish  the  effective  length  of  the  weir  by  Q.iH. 

FRANCIS'  method  of  correcting  for  velocity  of  approach 
differs  from  that  of  SMITH,  and  is  the  same  as  that  explained 
in  Art.  25.  The  head  h  causing  the  velocity  of  approach  is 
computed  in  the  usual  way,  and  then  the  formulas  are  written, 
for  weirs  without  end  contractions, 

*  =  3.33*K#+W--W];     .  ',    .    ;     (35)' 
and  f9r  weirs  with  end  contractions, 

?=3-33(*-o.2#)[(# +*)«-#]•  .    .    ,    (36)' 

It  is  necessary  that  this  method  of  introducing  the  velocity  of 
approach  should  be  strictly  observed,  since  the  mean  number 
3.33  was  deduced  for  this  form  of  expression. 

•  It  is  seen  that  the  number  3.33  is  c  .  f  V2g%  where  c  is  the 
true  coefficient  of  discharge.  The  88  experiments  from  which 
this  mean  value  was  deduced  show  that  the  coefficient  3.33 
actually  ranged  from  3.30  to  3.36,  so  that  by  its  use  an  error 
of  one  per  cent  in  the  computed  discharge  may  occur.  When 
such  an  error  is  of  no  importance  the  formula  may  be  safely 
used  for  weirs  longer  than  4  feet  and  heads  greater  than  0.4 
feet. 

Prob.  73.  Find  by  FRANCIS'  formulas  the  discharge  when 
B  =  7  feet,  b  =  4  feet,  H  —  0.457  feet>  an<3  G  =  1.5  feet,  the 
weir  being  one  with  end  contractions. 


FLO W  OF    WATER   OVER    WEIRS.  [CHAP.  V. 


ARTICLE  56.  SUBMERGED  WEIRS. 

When  the  water  on  the  down-stream  side  of  the  weir  is  al- 
lowed to  rise  higher  than  the  level  of  the  crest  the  weir  is  said 
to  be  submerged.  In  such  cases  an  entire  change  of  condition 
results,  and  the  preceding  formulas  are  inapplicable.  Let  H  be 
the  head  above  the  crest  measured  up  stream  from  the  weir  by 
the  hook  gauge  in  the  usual  manner,  and  let  H'  be  the  head 
above  the  crest  of  the  water  down  stream  from  the  weir  meas- 
ured by  a  second  hook  gauge.  If  H  be  constant,  the  discharge 

is  uninfluenced  until  the  lower  water 
rises  to  the  level  of  the  crest,  provided 
that  free  access  of  air  is  allowed  be- 
neath the  descending  sheet  of  water. 
But  as  soon  as  it  rises  slightly  above 
the  crest  so  that  H'  has  small  values, 

the  contraction  is  suppressed  and  the  discharge  hence  increased. 
As  H'  increases,  however,  the  discharge  diminishes  until  it  be- 
comes zero  when  H'  equals  H.  Submerged  weirs  cannot  be 
relied  upon  to  give  precise  measurements  of  discharge  on 
account  of  the  lack  of  experimental  knowledge  regarding  them, 
and  should  hence  always  be  avoided  if  possible. 

The  following  method  for  estimating  the  discharge  over 
submerged  weirs  without  end  contractions  is  taken  from  the 
discussion  given  by  HERSCHEL*  of  the  experiments  made  by 
FRANCIS  and  by  FTELEY  and  STEARNS.  The  observed  head  H 
is  first  multiplied  by  a  number  n,  which  depends  upon  the 
ratio  of  H'  to  H,  and  then  the  discharge  is  to  be  found  by  the 
formula 


transactions  American  Society  of  Civil  Engineers,  1885,  vol.  xiv.  p.  194. 


ART.  56.]  SUBMERGED    WEIRS. 

The  -values  of  n  are  given  in  the  following  table : 
TABLE   XII.    SUBMERGED   WEIRS. 


117 


H' 

Jr 

n 

H> 
H 

n 

H1 

a 

n 

H 
~H 

n 

0.00 

I.OOO 

0.18 

0.989 

0.38 

0-935 

0.58 

0.856 

.01 

1.004 

.20 

0.985 

.40 

0.929 

.60 

0.846 

.02 

1.  006 

.22 

0.980 

.42 

0.922 

.62 

0.836 

.04 

I.OO7 

.24 

0.975 

•44 

0.915 

.64 

0.824 

.06 

1.007 

.26 

0.970 

.46 

0.908 

.66 

0.813 

.08 

I.  OO6 

.28 

0.964 

.48 

0.900 

.70 

0.787 

.10 

1.005 

•30 

0-959 

•50 

0.892 

•75 

0.750  ' 

.12 

I.OO2 

•32 

0-953 

•52 

0.884 

.80 

0.703 

.14 

0.998 

•  34 

0.947 

•54 

0.875 

.90 

0-574 

.16 

0.994 

•36 

0.941 

•56 

0.866 

I.OO 

O.QOO 

The  numbers  in  this  table  are  liable  to  a  probable  error  of 
about  one  unit  in  the  second  decimal  place  when  H'  is  less  than 
O.2//,  and  to  greater  errors  in  the  remainder  of  the  table,  those 
values  of  n  less  than  0.70  being  in  particular  uncertain.  This 
discussion  shows  that  H'  may  be  nearly  one-fifth  of  H  without 
affecting  the  discharge  more  than  two  per  cent. 

A  rational  formula  for  the  discharge  over  submerged  weirs 
may  be  deduced  in  the  following  manner.  The  theoretic  dis- 
charge may  be  regarded  as  composed  of  two  portions,  one 
through  the  upper  part  H  —  H'  ,  and  the  other  through  the 
lower  part  H'  '.  The  portion  through  the  upper  part  is  given 
by  the  usual  weir  formula,  H  —  H'  being  the  head,  or 


and  that  through  the  lower  part  is  given  by  the  formula  for  a 
submerged  orifice  (Art.  42),  in  which  b  is  the  breadth,  H'  the 
height,  and  H  —  H'  the  effective  head,  or 


,  =  bH'  V2g(H  -  H'). 


Il8  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

The  addition  of  these  gives  the  total  theoretic  discharge, 

Q  =  I  \^rb(H-HJ+  V^bH'(H  -  HJ. 
This  may  be  put  into  the  more  convenient  form, 


The  actual  discharge  per  second  may  now  be  written, 

in  which  c  is  the  coefficient  of  discharge. 

FTELEY  and  STEARNS  adopt  the  above  formula  for  the  dis- 
charge, or  placing  m  for  c .  f  V2g,  they  write,* 

and  from  their  experiments  deduce  the  following  values  of  m : 

TTf 

For   -=  =  0.00      0.04      0.08      0.12      0.16      0.2        0.3 
rl 

*«  =  3-33       3-35       3-37      3-35       3-32       3-28       3.21 

TTf 

For  -77-  =  0.4        0.5        0.6        0.7        0.8        0.9        i.o 

£1 

m  =  3.i$       3.11       3-09       3-09       3-12       3.19       3.33 

These  are  for  suppressed  weirs  ;  for  contracted  weirs  few  or  no 
experiments  are  on  record. 

In  what  has  thus  far  been  said  velocity  of  approach  has  not 
been  considered.  This  may  be  taken  into  account  in  the  usual 
way  by  determining  the  velocity-head  //,  and  thus  correcting 
H.  In  strictness  the  velocity  of  departure  in  the  tail  bay  below 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  xii.  p.  103. 


ART.  57-]  ROUNDED  AND    WIDE   CRESTS.  1 19 

the  weir  should  be  regarded,  and  its  head  h'  be  applied  to  H' 
But  it  is  unnecessary,  on  account  of  the  limited  use  of  sub- 
merged weirs,  and  the  consequent  lack  of  experimental  data,  to 
develop  this  branch  of  the  subject.  What  has  been  given 
above  will  enable  a  probable  estimate  to  be  made  of  the  dis- 
charge in  cases  where  the  water  accidentally  rises  above  the 
crest,  and  further  than  this  the  use  of  submerged  weirs  canriot 
be  recommended. 

Prob.  74.  Compute  by  two  methods  the  discharge  over  a 
submerged  weir  when  b  =  8,  H  =  0.46,  and  H'  =  0.22  feet. 


ARTICLE  57.  ROUNDED  AND  WIDE  CRESTS. 

When  the  inner  edge  of  the  crest  of  a  weir  is  rounded,  as  at 
A  in  Fig.  34,  the  discharge  is  materially  increased  as  in  the  case  of 
orifices  (Art.  44),  or  rather  the  coefficients  of  discharge  become 
much  larger  than  those  given 
for  the  standard  sharp  crests. 
The  degree  of  rounding  influ- 
ences so  much  the  amount  of 
increase  that  no  definite  values  FlG-  34- 

can  be  stated,  and  the  subject  is  here  merely  mentioned  in  order 
to  emphasize  the  fact  that  a  rounded  inner  edge  is  always  a 
source  of  error.  If  the  radius  of  the  rounded  edge  is  small, 
the  sheet  of  escaping  water  leaves  it  at  a  point  below  the  top 
(a  in  the  figure),  which  has  the  practical  effect  of  increasing  the 
measured  head  by  a  constant  quantity.  The  experiments  of 
FTELEY  and  STEARNS  show  that  when  the  radius  is  less  than 
one-half  an  inch,  the  discharge  can  be  computed  from  the  usual 
weir  formula,  seven-tenths  of  the  radius  being  first  added  to 
the  measured  head  H. 

Two  wide-crested  weirs  with  square  inner  corners  are  shown 
in  Fig.  34,  the  one  at  B  being  of  sufficient  width  so  that  the 


120 


FLO W  OF    WATER    OVER    WEIRS. 


[CHAP.  V. 


descending  sheet  may  just  touch  the  outer  edge,  causing  the 
flow  to  be  more  or  less  disturbed,  while  that  at  C  has  the  sheet 
adhering  to  the  crest  for  some  distance.  In  both  cases  the 
crest  contraction  occurs,  although  water  instead  of  air  may  fill 
the  space  above  the  inner  corner.  For/?  the  discharge  maybe 
equal  to  or  greater  than  that  of  the  standard  weir  having  the 
same  head  H,  depending  upon  whether  the  air  has  or  has  not 
free  access  beneath  the  sheet  in  the  space  above  the  crest.  For 
C  the  discharge  is  always  less  than  that  of  the  standard  weir 
with  sharp  crest. 

The  following  table  is  an  abstract  from  the  results  obtained 
by  FTELEY  and  STEARNS,*  and  gives  the  corrections  in  feet  to 
be  subtracted  from  the  depths  on  a  wide  crest,  like  C  in  Fig. 
34,  in  order  to  obtain  the  depths  on  a  standard  sharp-crested 
weir  which  will  discharge  an  equal  volume  of  water. 

TABLE   XIII.    CORRECTIONS    FOR   WIDE   CRESTS. 


Head 
on  wide 
crest. 
Feet. 

Width  of  crest  in  inches. 

2 

4 

6 

8 

IO 

12 

24 

0.05 

0.010 

O.OOg 

O.OO9 

0.009 

0.009 

O.OOg 

0.009 

.IO 

.Ol6 

.018 

.017 

.017 

.017 

.017 

.017 

.20 

.012 

.029 

.031 

.032 

•  033 

•033 

•034 

•30 

.030 

.041 

.045 

.047 

.048 

•  050 

.40 

.022 

•  045 

.055 

.060    |       .062 

.066 

•50 

.OO6 

.041 

.060 

.069           .074 

.082 

.60 

.031 

.059 

.075 

.083 

.097 

.70 

.017 

.052 

.075 

.089 

.112 

.80 

.000 

.040 

.071 

.091 

.125 

.90 

.027 

.062 

.089 

•137 

1.  00 

.Oil 

.050 

.082 

.149 

1.20 

.021 

.061 

.168 

I.4O 

.032 

.180 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  xii.  96. 


ART.  58].  WASTE    WEIRS  AND  DAMS.  121 

These  results  were  obtained  by  passing  a  constant  volume 
of  water  over  a  standard  weir  and  measuring  the  head  H  on  the 
crest ;  a  piece  of  timber  was  then  brought  into  place  on  the 
lower  side  of  the  crest  and  secured  by  fastenings,  thus  forming 
the  wide  crest ;  and  the  head  H  being  again  measured,  the  in- 
crease of  depth  was  thus  obtained.  This  being  repeated  for 
different  constant  volumes  the  results  were  plotted  and  mean 
curves  drawn,  from  which  the  table  was  derived.  The  weir 
used  was  without  end  contractions,  and  to  such  only  the  con- 
clusions apply  with  precision.  For  weirs  with  end  contractions 
where  the  air  has  free  access  under  the  sheet  at  the  ends  the 
discharge  is  probably  greater. 

Prob.  75.  Compute  the  discharge  over  a  crest  1.5  feet  wide 
for  a  weir  10  feet  long  when  the  head  is  0.850  feet,  and  show 
that  the  discharge  is  about  19  per  cent  less  than  that  over  a 
standard  sharp-crested  weir  under  the  same  head. 

ARTICLE  58.  WASTE  WEIRS  AND  DAMS. 

Waste  weirs  are  constructed  at  the  sides  of  canals  and 
reservoirs  in  order  to  allow  surplus  water  to  escape.  They  are 
usually  made  with  wide  crests,  the  inner  approach  to  which  is 
inclined,  and  the  discharge  is  received  upon  an  apron  of  timber 
or  masonry.  The  flow  over  these  wide-crested  weirs  is  always 


FIG.  35. 


much  less  than  for  equal  depths  on  standard  weirs,  and  for 
narrow  crests  the  diminution  may  be  approximately  estimated 
by  the  use  of  the  table  in  the  preceding  article.  When 
the  crest  is  about  3  feet  wide,  and  level,  with  a  rising  slope 


122  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

to    its  inner  edge,  and  the  end    contractions  are   suppressed, 
the  following  formula,  deduced  by  FRANCIS,  may  be  applied, 

q  — 

in  which  b  and  H  are  to  be  taken  in  feet,  and  q  is  in  cubic  feet 
per  second. 

In  constructing  a  waste  weir  the  discharge  q  is  generally 
known  or  assumed,  and  it  is  required  to  determine  b  and  H. 
The  latter  being  taken  at  I,  2,  or  3  feet,  as  may  be  judged  safe 
and  proper,  b  is  found  by 


3.oi//153 

If,  for  example,  q  be  87  cubic  feet  per  second,  and  H  be  taken 
as  2  feet,  then 


=  log  87  —  log  3.01  —  i. 53  log  2, 
from  which 

log  b  —  1.0004, 

whence  b  =  10.0  feet.     If,  however,  H  be  taken  as  I  foot,  b  is 
required  to  be  nearly  30  feet. 

The  ordinary  weir  formula  may  be  also  used  for  waste-weir 
calculations  with  results  differing  but  little  from  those  obtained 
by  the  above  expression.  Or  using  the  approximate  general 
expression  from  Art.  55, 


b  = 


3-33^* 


In  this,  if  q  be  87  cubic  feet  per  second,  and  H  be  2  feet,  the 
value  of  b  is  found  to  be  9.24  feet.  Evidently  no  great  pre- 
cision is  needed  in  computing  the  length  of  a  waste  weir,  since 


ART.  58.] 


WASTE    WEIRS  AND  DAMS. 


123 


it  is  difficult  to  determine  the  exact  discharge  which  is  to  pass 
over  it,  and  ample  allowance  must  be  made  for  unusual  rains 
or  floods. 

When  a  dam  is  built  across  a  stream  it  is  often  important 
to  arrange  its  height  so  that 
the  water  level  may  stand 

In 


at  a  certain  elevation. 
Fig.  36  the  line  CC  repre- 
sents the  surface  of  the 
stream  before  the  construc- 
tion of  the  dam,  the  depth 
of  water  being  D,  and  it  is 
required  to  find  the  height 
of  the  dam  G,  so  that  the 
surface  may  be  raised  the 
distanced.  If  the  crest  be  FlG- 36- 

not  submerged,  as  in  the  first  diagram, 

G  =  D  +  df  -  H. 

In  this  H  is  to  be  inserted  in  terms  of  the  discharge  q,  or  the 
length  b  is  to  be  determined  as  above  for  an  assumed  value  of 
H.  For  the  former  method, 


in  which  b  may  be  width  of  the  stream  or  less,  as  the  design 
requires.  If  G,  D,  q,  and  b  be  given,  this  formula  may  be  used 
to  compute  d1 '. 

If  the  height  of  the  dam  is  small,  as  in  the  second  diagram 
of  Fig.  36,  the  crest  is  submerged,  and  the  last  formula  will  not 
apply.  For  this  case 


H  = 


Hf  =  D- 


124  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

and  inserting  these  heads  in  the  formula  (37)',  and  solving  for 
G,  the  following  result  is  found : 

r       n       u>  2q 

(jr   "=•  JJ  -j-  ^U      — 


In  this  formula  m  lies  between  3.09  and  3.37,  depending  on  the 
value  of  the  ratio  H '  -~  H,  and  accordingly  a  tentative  method 
of  solution  must  be  adopted.  For  example,  let  D  =  4  feet, 
d'  =  i  foot,  b  =  50  feet,  and  q  —  400  cubic  feet  per  second ; 
then,  assuming  m  as  3.33, 

G  =  4  +0.67  —  1.6  =  3.1  feet. 

Now  H=  4  +  i  —  3.1  =  1.9  feet,  and  Hf  =  4  —  3.1  =  0.9,  so 
that  the  ratio  H '  -f-  H  —  0.47,  and  hence  from  Art.  56  the 
value  of  m  is  about  3.12.  Using  this,  the  value  of  G  is  now 
computed  to  be  2.96  feet,  which  gives  H  =  2.04  feet,  and 
H'  =  1.04  feet,  and  H'  -\-  H  =  0.5,  which  indicates  that  no 
further  variation  in  m  will  be  found.  Accordingly  2.96  feet  is 
the  required  height  of  the  submerged  dam. 

Prob.  76.  If  150  cubic  feet  per  second  Bow  over  a  waste 
weir  20  feet  long,  find  the  depth  of  water  on  the  crest. 

Prob.  77.  A  stream  4  feet  deep  which  delivers  150  cubic  feet 
per  second  is  to  be  dammed  so  as  to  raise  the  water  6  feet 
higher.  Find  the  height  of  the  dam  when  the  length  of  the 
overflow  is  12  feet. 


ARTICLE  59.  THE  SURFACE  CURVE. 

The  surface  of  the  water  above  a  weir  assumes  during  the 
flow  a  curve  whose  equation  is  not  known,  but  some  of  the 
laws  which  govern  it  maybe  deduced  in  the  following  manner: 
Let  H  be  the  head  above  the  level  of  the  crest  measured  in 


ART.  59.] 


THE   SURFACE   CURVE. 


125 


perfectly  level  water  at  some  distance  back  of  the  weir,  and  let 
d  be  the  depression  or  drop  of  the  curve 
below  this  level  in  the  plane  of  the  weir 
(Fig.  37).  The  discharge  per  second  q 
can  be  expressed  in  terms  of  H  and  d  by 
formula  (11)'  of  Art.  25  by  placing  H  for 
h^  and  d  for  /^  .  This,  multiplied  by  a 
F*G-  37.  coefficient  k,  gives,  if  velocity  of  approach 

be  neglected,  the  formula 

q  =  k.\*/2g.b(H*  —  d?). 

This  expression,  it  may  be  remarked,  is  the  true  weir  formula, 
and  only  the  practical  difficulties  of  measuring  d  prevent  its 
use. 

From  this  formula  the  value  of  the  drop  d  in  the  plane  of 
the  weir  is  found  to  be 


2kb  \2g 

Let  B  be  the  breadth  of  the  feeding  canal,  G  its  depth  below 
the  crest,  and  v  the  mean  velocity  of  approach  ;  then 

q  =  B(G  +  H)v. 

3  ^ 

Inserting  this  in  the  equation,  replacing  —r  by  m,  and  —  -=  by 

2.R  \/  2g 

its  value  fa,  where  h  is  the  velocity-head  corresponding  to  v, 
the  formula  becomes 


(38) 


which  is  an  expression  for  the  drop  of  the  curve  in  terms  of  the 
dimensions  of  the  feeding  canal  and  weir,  and  the  heads  H 
and  h. 


126  FLOW  OF    WATER   OVER    WEIRS.  [CHAP.  V. 

The  approximate  value  of  the  coefficient  m  is  about  2.2, 
but  precise  values  of  d  cannot  be  computed  unless  m  and  H 
are  known  with  accuracy.  The  formula,  however,  serves  to 
exemplify  the  laws  which  govern  the  drop  of  the  curve  in  the 
plane  of  the  weir.  It  shows  that  the  drop  increases  with  the 
head  on  the  crest  and  with  the  length  of  a  contracted  weir,  that 
it  decreases  with  the  breadth  and  depth  of  the  feeding  canal> 
and  that  it  decreases  with  the  velocity  of  approach.  It  also 
shows  for  suppressed  weirs,  where  B  —  b,  that  the  drop  is  inde- 
pendent of  the  length  of  the  weir.  All  of  these  laws  except 
the  last  have  been  previously  deduced  by  the  discussion  of 
experiments. 

Prob.  78.  Discuss  the  above  formula  when  H  =  o ;  also 
when  h  =  o. 


ARTICLE  60.  TRIANGULAR  WEIRS. 

Triangular  ribtches  are  used  but  little,  as  in  general  they  are 
only  convenient  when  the  quantity  of  water  to  be  measured  is 
small.  Such  a  notch  when  used  as  a  weir  must  have  sharp 
inner  corners,  so  that  the  stream  may  be  fully  contracted,  and 
the  sides  should  have  equal  slopes.  The  angle  at  the  lower 
vertex  should  be  a  right  angle,  as  this  is  the  only  case  for  which 
coefficients  are  known  with  precision.  The  depth  of  water 
above  this  lower  vertex  is  to  be  measured  by  a  hook  gauge  in 
the  usual  manner  at  a  point  several  feet  up  stream  from  the 
notch. 

In  Art.  23  is  deduced  a  formula  for  the  theoretic  discharge 
through  a  triangular  notch.  Making  the  angle  at  the  vertex  a 
right  angle,  and  applying  a  coefficient,  the  theoretic  discharge 
per  second  is 

Q  =  c.  -ft 


ART.  60.]  TRIANGULAR    WEIRS.  1 27 

in  which  H  is  the  head  of  water  above  the  vertex.  If  velocity 
of  approach  exists,  H  may  be  increased  by  the  velocity-head  h 
as  for  rectangular  weirs. 

Experiments  made  by  THOMSON  *  indicate  that  the  coef- 
ficient c  varies  less  with  the  head  than  for  ordinary  weirs ;  this 
in  fact  was  anticipated,  since  the  sections  of  the  stream  are 
similar  in  a  triangular  notch  for  all  values  of  H,  and  hence 
the  influence  of  the  contractions  in  diminishing  the  discharge 
should  be  nearly  constant.  As  the  result  of  his  experiments 
the  mean  value  of  c  for  heads  between  0.2  and  0.8  feet  may  be 
taken  as  0.592.  If,  further,  8.02  be  put  for  \^2gy  the  discharge 
in  cubic  feet  per  second  may  be  written 


q  = 
in  which  H  must  be  expressed  in  feet. 

Prob.  79.  Find  the  size  of  a  triangular  notch  to  discharge 
about  50  cubic  feet  per  second.  Also  the  size  of  a  rectangular 
weir  to  discharge  the  same  quantity,  when  the  head  is  1.5  feete 

*  British  Association  Report,  1858,  p.  133. 


128  FLO  W   THROUGH   TUBES.  [CHAP.  VI 


CHAPTER  VI. 
FLOW  THROUGH  TUBES. 

ARTICLE  61.  THE  STANDARD  SHORT  TUBE. 

A  standard  tube  is  a  very  short  pipe,  whose  length  is 
about  three  times  its  diameter,  or  of  sufficient  length  so  that 
the  escaping  jet  just  fills  its  outer  end, 
and  there  issues  without  contraction. 
The  inner  end  of  the  tube  is  placed  flush 
with  the  inner  side  of  the  reservoir,  and 
is  to  be  a  sharp,  definite  corner,  like  that 
of  the  standard  orifice  (Art.  34).  FIG.  3s. 

The  phenomena  of  flow  through  such  a  tube  are  similar  in 
some  respects  to  those  of  the  flow  from  the  standard  orifice, 
but  the  discharge  is  much  greater.  By  observations  with  glass 
tubes  it  is  found  that  the  contraction  of  the  jet  occurs  as  in  the 
orifice,  although  agitation  of  the  water  or  a  shock  upon  the 
tube  is  apt  to  apparently  destroy  it,  and  cause  the  entire  length 
to  be  filled.  If,  however,  holes  be  bored  in  the  tube  near  its 
inner  end,  water  does  not  flow  out,  but  air  enters,  showing  that 
a  negative  pressure  exists. 

Since  the  issuing  jet  entirely  fills  the  outer  end  of  the  tube, 
the  coefficient  of  contraction  for  that  section  is  unity  (Art.  35), 
and  hence  the  coefficient  of  velocity  equals  the  coefficient  of 
discharge  (Art.  37).  Numerous  experiments  by  VENTURI, 
BOSSUT,  CASTEL,  and  others,  give  the  following  as  a  mean 
value  for  the  standard  tube : 

c  =  0.82. 


ART.  61.] 


THE   STANDARD  SHORT   TUBE. 


I29 


This  value,  however,  ranges  from  0.83  for  low  heads  and  small 
tubes  to  0.80  for  high  heads  and  large  tubes,  its  law  of  varia- 
tion being  probably  the  same  as  for  orifices  (Art.  38),  although 
experiments  are  wanting  from  which  to  state  definite  values  in 
the  form  of  a  table. 

A  standard  orifice  gives  on  the  average  about  61  per  cent 
of  the  theoretic  discharge,  but  by  the  addition  of  a  tube  this 
may  be  increased  to  82  per  cent.  The  effective  energy  of  the 
jet  from  the  tube  is,  however,  much  less  than  that  from  the 
orifice.  For,  let  v  be  the  velocity  and  h  the  head,  then  (Art. 
36)  for  the  orifice 

v  =  0.98  V2gh,  whence  —  =  0.96^5  ; 
and  similarly  for  the  tube, 


v  =  0.82  ^2gh,  whence  —  =  0.67/2. 


Accordingly,  the  effective  energy  of  the  stream  from  the  orifice 

is  96  per  cent  of  the  theoretic 

energy,    while     that    of     the 

stream  from  the  tube  is  only 

67  per   cent.      Or   if  jets  be 

directed      vertically      upward 

from  a  standard  orifice  and  a 

standard  tube,  as   in  Fig.  39, 

that  from  the  former  rises  to 

the   height  0.96/2,  while   that 

from  the    latter  rises   to   the 

height  o.6;//,  where  h  is   the  FlG'  39> 

head  from  the  level  of  water  AB  in  the  reservoir  to  the  point 

of  exit. 


130  FLOW   THROUGH   TUBES.  [CHAP.  VL 

The  standard  tube  is  not  used  for  the  measurement  of  water, 
as  this  can  be  done  with  greater  precision  and  convenience  by 
orifices.  It  is  important,  however,  to  know  the  general  laws  of 
flow  which  have  here  been  set  forth,  as  a  starting  point  in  the 
theory  of  pipes,  and  for  other  purposes.  The  fact  that  the  tube 
gives  a  greater  discharge  than  an  orifice  is  an  interesting  one, 
and  the  reason  for  this  will  be  explained  in  Art.  67. 

Prob.  80.  Compare  the  effective  horse-power  of  the  streams 
from  a  standard  orifice  and  tube,  the  diameter  of  each  being 
4  inches  and  the  head  25  feet. 

ARTICLE  62.  CONICAL  CONVERGING  TUBES. 

Conical  converging  tubes  are  used  when  it  is  desired  to 
obtain  a  high  efficiency  in  the  energy  of  the  stream  of  water. 

At  A  is  shown  a  simple  con- 
verging tube,  consisting  of  a 
frustum  of  a  cone,  and  at  B 
is  a  similar  frustum,  provided 
with  a  cylindrical  tip.  The 
FlG-  40.  proportions  of  these  converg- 

ing tubes,  or  mouthpieces,  vary  somewhat  in  practice,  but  the 
cylindrical  tip  when  employed  is  of  a  length  equal  to  about 
2\  times  its  inner  diameter,  while  the  conical  part  is  eight  or 
ten  times  the  length  of  that  diameter,  the  angle  at  the  vertex 
of  the  cone  being  between  10  and  20  degrees. 

The  stream  from  a  conical  converging  tube  like  A  suffers  a 
contraction  at  some  distance  beyond  the  end.  The  coefficient 
of  discharge  is  higher  than  that  of  the  standard  tube,  being 
generally  between  0.85  and  0.95,  while  the  coefficient  of  velocity 
is  higher  still.  Experiments  made  by  D'AUBUISSON  and  CASTEL 
on  conical  converging  tubes  0.04  meters  long  and  0.0155  meters 
in  diameter  at  the  small  end,  under  a  head  of  3  meters,  give 


ART.  62.] 


CONICAL    CONVERGING    TUBES. 


the  following  results  for  the  coefficients  of  discharge  and 
velocity,  the  former  being  determined  by  measuring  the  actual 
discharge  (Art.  37),  and  the  latter  by  the  range  of  the  jet  (Art. 
36).  The  coefficient  of  contraction,  as  computed  from  these, 
is  given  in  the  last  column ;  and  this  applies  to  the  jet  at  the 
smallest  section,  some  distance  beyond  the  end  of  the  tube. 

TABLE  XIV.     COEFFICIENTS   FOR  CONICAL  TUBES. 


Angle  of  Cone. 

Discharge 
c. 

Velocity 

?!• 

Contraction 
c'. 

0°     00' 

0.829 

0.829 

1.  00 

I        36 

0.866 

0.867 

4      10 

0.912 

0.910 

7       52 

0.930 

0.932 

0.998 

10         20 

0.938 

0.951 

0.986 

13         24 

0.946 

0.963 

0.983 

16      36 

0.938 

0.971 

0.966 

21         00 

0.919 

0.972 

0-945 

29         58 

0.895 

0-975 

0.918 

48         50 

0.847 

0.984 

0.861 

While  these  values  show  that  the  greatest  discharge  occurred 
for  an  angle  of  about  13^-  degrees,  they  also  indicate  that  the 
coefficient  of  velocity  increases  with  the  convergence  of  the 
cone,  becoming  about  equal  to  that  of  a  standard  orifice  for 
the  last  value.  Hence  the  table  seems  to  teach  that  a  conical 
frustum  is  not  the  best  form  for  a  mouthpiece  to  give  the 
greatest  velocity. 

Under  very  high  heads — over  300  feet — SMITH  found  the 
actual  discharge  to  agree  closely  with  the  theoretical,  or  the 
coefficient  of  discharge  was  nearly  i.o,  and  in  some  cases  slightly 
greater.*  His  tubes  were  about  0.9  feet  long,  o.i  feet  in 


*  SMITH'S  Hydraulics,  p.  286. 


I32  FLO  W   THROUGH   TUBES.  [CHAP.  VI. 

diameter  at  the  small  end  and  0.35  feet  at  the  large  end,  the 
angle  of  convergence  being  17  degrees.  As  this  implies  a  con- 
traction of  the  jet  beyond  the  end,  it  cannot  be  supposed  that 
the  coefficient  of  discharge  in  any  case  was  really  as  high  as 
his  experiments  indicate.  Under  these  high  heads  the  cylin- 
drical tip  applied  to  the  end  of  a  tube  produced  no  effect  on 
the  discharge,  the  jet  passing  through  without  touching  its 
surface. 

Prob.  81.  If  the  coefficient  of  discharge  is  0.98  and  the 
coefficient  of  velocity  0.995,  compute  the  coefficient  of  con- 
traction. 

ARTICLE  63.  NOZZLES. 

For  fire  service  two  forms  of  nozzles  are  in  use.  The  smooth 
nozzle  is  essentially  a  conical  tube  like  A  in  Fig.  40,  the  larger 

end  being  attached  to  a  hose,  but  it  is 
HP  often  provided  with  a  cylindrical  tip 
and  sometimes  the  inner  end  is  curved 
as  seen  in  the  upper  diagram  of  Fig.  41. 
The  ring  nozzle  is  a  conical  tube  having 
an  orifice  whose  diameter  is  slightly 
smaller  than  that  of  the  end  of  the  tube. 
The  experiments  of  FREEMAN  show 

that  the  mean  coefficient  of  discharge  is  about  0.97  for  the 
smooth  nozzle  and  about  0.74  for  the  ring  nozzle.*  They  also 
seem  to  indicate  that  the  simple  cone  has  a  higher  discharge 
than  any  form  of  curved  nozzle. 

The  following  table  contains  approximate  heights  to  which 
jets  may  be  thrown  by  nozzles  according  to  the  investigations 
of  BOX  and  SHEDD,  as  quoted  in  the  tables  of  ELLIS,  f 

*  FREEMAN,  Experiments  relating  to  the  Hydraulics  of  Fire  Streams.  Trans- 
actions American  Society  of  Civil  Engineers,  1889. 

f  G.  A.  ELLIS,  Work  done  by  and  Power  required  for  Fire  Streams,  Spring- 
field, Mass.,  1878. 


ART.  63.] 


NOZZLES. 


This  gives  the  vertical  heights  in  feet  reached  by  jets  under 
different  conditions,  the  first  column  containing  the  effective 
pressure  at  the  entrance  to  the  nozzle,  and  the  second  the  cor- 
responding effective  head. 

TABLE  XV.     VERTICAL  HEIGHTS  OF  JETS   FROM   NOZZLES. 


Pressure  in 
Pounds  per 
'Square  Inch. 

Head  in 
Feet. 

From  i-inch  Nozzle. 

From  i|-inch  Nozzle. 

From  ij-inch  Nozzle. 

Smooth. 

Ring. 

Smooth. 

Ring. 

Smooth. 

Ring. 

10 

23 

22 

22 

22 

22 

23 

22 

20 

46 

43 

42 

43 

43 

43 

43 

30 

69 

62 

61 

63 

62 

63 

63 

40 

92 

79 

78 

81 

79 

82 

80 

50 

H5 

94 

92 

97 

94 

99 

95 

60 

138 

108 

104 

112 

108 

H5 

no 

70 

161 

121 

H5 

125 

121 

129 

123 

80 

184 

131 

124 

137 

131 

142 

135 

QO 

207 

140 

132 

I48 

141 

154 

146 

100 

230 

148 

136 

157 

149 

164 

155 

The  effective  head  at  the  entrance  of  a  nozzle  is  the  pres- 
sure-head plus  the  velocity-head  (Art.  27).  Let  d  be  the  diam- 
eter of  the  pipe,  and  dl  that  of  the  outlet  end  of  the  nozzle, 
and  v  and  v^  the  corresponding  velocities.  Let  /^  be  pressure- 
head  at  the  entrance  ;  then  the  effective  head  is 


and  the  velocity  of  discharge  is 


— 


134  FLOW  THROUGH   TUBES.  [CHAP.  VI. 

Now  if  /jj  and  v  are  known,  vl  may  be  computed.     But  if  hy  is 
alone  known,  this  equation  may  be  written, 


also  (Art.  19) 

v&  =  vd\ 

Inserting  in  the  first  expression  the  value  of  v  taken  from  the 
second,  and  solving  for  vl  ,  gives 


Here  the  last  term  in  the  denominator  shows  the  effect  of  the 
velocity  of  approach  in  the  pipe,  and  if  ct  =  i  it  agrees  with 
the  theoretic  expression  deduced  in  Art.  25.  In  order  to  use 
this  formula  h^  must  be  found  by  observation;  one  way  of 
doing  this  is  by  a  pressure  gauge  at  the  end  of  the  pipe 
which  reads  the  pressure  pl  in  pounds  per  square  inch  ;  then 
h^  =  2.304^  (Art.  9).  If  dl  be  small  compared  with  d  the 
formula  reduces  to  v  = 


The  question  as  to  the  proper  form  of  curve  for  a  nozzle  in 
order  that  the  velocity  may  be  a  maximum  is  an  interesting 
one.  It  is  thought  that  this  form  is  similar  to  that  of  a  jet 
which  rises  vertically  in  a  vacuum  from  an  orifice,  the  sections 
increasing  in  size  as  the  velocity  diminishes.  In  the  case  of 
the  jet  the  energy  of  the  stream  is  expended  against  the  con- 
stant force  of  gravity  ;  in  the  nozzle  the  constant  pressure  at 
the  entrance  is  converted  into  the  energy  of  the  stream.  In 
the  jet  the  velocity  is  retarded  according  to  a  certain  law,  and 
hence  it  seems  that  in  the  nozzle  the  stream  should  be  acceler- 
ated according  to  the  same  law.  The  height  of  the  perfect  jet 
is  the  same  as  the  head  k  under  which  it  issues  from  an  orifice  ; 


ART   03]  NOZZLES.  135 

the  length  of  the  nozzle  /,  however,  must  be  short  in  order  to 
avoid  frictional  resistances. 

Let  d^  be  the  diameter  of  the  jet  at  the  section  where  the 
velocity  is  V2gk.  At  any  distance  x  above  this  point  let  the 
diameter  be  y  ;  the  velocity  in  this  section  is  V2g(h  —  x).  Then, 
since  the  areas  of  the  sections  are  inversely  as  the  velocities, 


d?        V2g(h  -  x) 
and  from  this  the  value  of  y  is 


which  gives  the  law  of  variation  between  the  diameters  of  the 
different  sections.  In  this  formula  y  =  oo  when  x  =  h,  which 
should  be  the  case*for  a  theoretically  perfect  jet  rising  from  an 
orifice  to  the  level  of  the  reservoir  where  all  its  particles  are 
without  velocity.  Now  if  L  be  the  length  of  a  nozzle  at  whose 
entrance  the  water  has  no  velocity,  and  dl  the  diameter  at  the 
small  end,  the  diameter  at  any  distance  from  that  end  is 


-  •  •  •  •  •  (4o) 

To  show  the  form  of  profile  the  following  values  of  y  for  cor- 
responding values  of  x  are  given  : 

For  ;t;  =:  o.iZ,        0.3         0.5         0.7         O.8         0.9         0.99 
1.09       1.19       1.35        1.50       1.78       3.16 


In  practice  the  nozzle  is  attached  to  a  hose  or  pipe  whose 
diameter  is  d,  so  that  the  water  enters  with  a  certain  velocity. 
Let  /  be  the  length  of  the  nozzle  in  this  case  ;  then  in  the 
above  equation  y  equals  d  when  x  equals  /,  and 


136  FLOW   THROUGH   TUBES.  [CHAP.  VI. 

To  determine  the  'diameter  y  at  any  distance  x  from  the  small 
end,  L  may  be  eliminated  from  the  two  equations,  giving  the 
formula 


from  which  y  may  be  computed  for  given  values  of  x,  the 
diameters  d^  and  d  being  first  assumed.  The  value  of  the 
length  /  in  practice  is  often  between  6dl  and  iodl  ,  while  d  is 
about  3^  .  The  best  relations  between  /,  d^  ,  and  d  depend 
upon  frictional  resistances,  which  have  here  been  neglected,  and 
upon  considerations  of  convenience  .  The  following  "are  values 
of  y  for  corresponding  values  of  x  when  d  =  3^  : 

For  x  —   o.i  /       0.3        0.5         0.7        0.8        0.9         i.o 
y=i.o$d      1.09       1.19       1.34       1.48       1.74       3.0 

These  are  seen  to  closely  agree  with  the  values  deduced  above 
for  the  nozzle  whose  diameter  d  is  infinite,  and  accordingly  the 
equation  for  that  case  may  be  taken  as  a  close  expression  for 
the  theoretically  perfect  nozzle  whenever  d  is  greater  than  $dr 
The  formula  (40)  has  also  been  deduced  by  NAGLE  from  the 
principle  that  the  velocity  in  the  nozzle  should  be  uniformly 
accelerated.* 

Prob.  82.  Compute  the  coefficient  of  velocity  for  a  nozzle 
whose  jet  rises  to  a  vertical  height  of  32  feet  when  the  effective 
pressure  at  the  entrance  is  15  pounds  per  square  inch. 

Prob.  83.  If  the  coefficient  of  velocity  is  0.98,  compute  the 
velocity  from  a  nozzle  of  I  inch  diameter  when  attached  to  a 
hose  of  2-J-  inches  diameter,  the  pressure  at  the  entrance,  as 
measured  by  a  gauge,  being  43.4  pounds  per  square  inch. 

Ans.  79.6  feet  per  second. 

*  Transactions  American  Society  of  Mechanical  Engineers,  1888. 


ART.  64.]  DIVERGING  AND   COMPOUND    TUBES. 


137 


FIG.  42. 


ARTICLE  64.  DIVERGING  AND  COMPOUND  TUBES. 

In  Fig.  42  is  shown  a  diverging  conical  tube  JBC,  and  two 
compound  tubes.  The  compound  tube  ABC  consists  of  two 
cones,  the  converging  one,  AB,  being  much  shorter  than  the 
diverging  one,  BC,  so  that  the 
shape  roughly  approximates  to 
the  form  of  the  contracted  jet 
which  issues  from  an  orifice  in 
a  thin  plate.  In  the  tube  AE 
the  curved  converging  part  AB 
closely  imitates  the  contracted 
jet,  and  BB  is  a  short  cylinder 
in  which  all  the  filaments  of 
the  stream  are  supposed  to 
move  in  lines  parallel  to  the 
axis  of  the  tube,  the  remaining 
part  being  a  frustum  of  a  cone.  The  converging  part  of  a 
compound  tube  is  often  called  a  mouthpiece,  and  the  diverging 
part  an  adjutage. 

Many  experiments  with  these  tubes  have  shown  the  interest- 
ing and  phenomenal  fact  that  the  discharge  and  the  velocity 
through  the  smallest  section,  B,  are  greater  than  those  due  to  the 
head ;  or,  in  other  words,  that  the  coefficients  of  discharge  and 
velocity  are  greater  than  unity.  One  of  the  first  to  notice  this 
was  BERNOULLI  in  1738,  who  found  c  —  1.08  for  a  diverging 
tube.  VENTURI  in  1791  experimented  on  such  tubes,  and 
showed  that  the  angle  of  the  diverging  part,  as  also  its  length, 
greatly  influenced  the  discharge.  He  concluded  that  c  would 
have  a  maximum  value  of  1.46  when  the  length  of  the  diverg- 
ing part  was  9  times  its  least  diameter,  the  angle  at  the  vertex 
of  the  cone  being  5°  06'.  EYTELWEIN  found  c  =  1.18  for.  a 
diverging  tube  like  BC  in  Fig.  42,  but  when  it  was  used  as  an 


138  FLO  W   THROUGH   TUBES.  [CHAP.  VI. 

adjutage  to  a  mouthpiece,  AB,  thus  forming  a  compound  tube 
ABC,  he  found  c  =  1.55. 

The  experiments  of  FRANCIS  in  1854  on  a  compound  tube 
like  ABODE  are  very  interesting.*  The  curve  of  the  converg- 
ing part  AB  was  a  cycloid,  BB  was  a  cylinder,  and  the  diameters 
at  A,  B,  etc.,  were 

A  =  1.4  feet,  £7  =  0.1454,  E  =  0.3209 

B  —  0.1018,  D  =  0.2339, 

The  piece  BB  was  o.i  feet  long,  and  the  others  each  I  foot; 
these  were  made  to  screw  together,  so  that  experiments  could 
be  made  on  different  lengths.  A  sixth  piece,  EF,  not  shown 
in  the  figure,  was  also  used,  which  was  a  prolongation  of  the 
diverging  cone,  its  largest  diameter  being  0.4085  feet.  The 
tubes  were  of  cast-iron,  and  quite  smooth.  The  flow  was 
measured  with  the  tubes  submerged,  and  the  effective  head 
varied  from  about  q.oi  to  1.5  feet.  Excluding  heads  less  than 
o.i  feet,  the  following  shows  the  range  in  value  of  the  coeffi- 
cients of  discharge  : 

c  for  Section  BB.  c  for  Outer  End. 

For  tube  AB.  0.80  to  0.94  0.80  to  0.94 

For  tube  AC,  1.43  to  1.59  0.70  to  0.78 

For  tube  AD,  1.98  to  2.16  0.37  to  0.41 

For  tube  AE,  2.08  to  2.43  0.21  to  0.24 

For  tube  AF,  2.05  to  2.42  0.13  to  0.15 

The  maximum  discharge  was  thus  found  to  occur  with  the 
tube  AE,  and  to  be  2.43  times  the  theoretic  discharge.  In 
general  the  coefficients  increased  with  the  heads,  the  value  2.08 
being  for  a  head  of  0.13  feet  and  2.43  for  a  head  of  1.36  feet; 
under  1.39  feet,  however,  c  was  found  to  be  2.26. 

The  value  of  g  at  Lowell,  Mass.,  where  these  experiments 

*  Lowell  Hydraulic  Experiments,  4th  Edition,  pp.  209-232. 


ART.  64.]          DIVERGING  AND   COMPOUND    TUBES.  139 

were  made,  is  about   32.162  feet  per  second.     Hence  under  a 
head  of  1.36  feet  the  theoretic  velocity  is 


=.  8.0202  1/1.36  =  9.36  feet  per  second, 
while  the  actual  velocity  in  the  section  BB  was 

v  =  2.43  X  9.36  =  22.74  feet  per  second. 
The  velocity-head  corresponding  to  this  is 

^  =  (2.43)V;  =  3.90/1. 

Therefore  the  flow  through  the  section  BB  was  that  due  to  a 
head  5.9  times  greater  than  the  actual  head  of  1.36  feet  ;  or,  in 
other  words,  the  energy  of  the  water  flowing  in  BB  was  5.9 
times  the  theoretic  energy.  Here,  apparently,  is  a  striking 
contradiction  of  the  fundamental  law  of  the  conservation  of 
energy. 

Under  high  heads  the  velocity  becomes  so  great  that  the 
jet  does  not  touch  the  sides  of  the  diverging  tube,  or  adjutage, 
and  hence  the  actual  may  not  exceed  the  theoretic  discharge. 
It  is  probable,  however,  that  if  the  tube  be  long  and  its  taper 
very  slight  an  increased  discharge  can  be  obtained  under 
a  high  head. 

The  explanation  of  the  phenomena  of  increased  velocity 
and  discharge  caused  by  these  tubes  is  simple.  It  is  due  to 
the  occurrence  of  a  partial  vacuum  near  the  inner  end  of  the 
adjutage  BC.  The  pressure  of  the  atmosphere  on  the  water 
in  the  reservoir  thus  increases  the  hydrostatic  pressure  due  to 
the  head,  and  the  increased  flow  results.  The  energy  at  the 
smallest  section  is  accordingly  higher  than  the  theoretic 
energy,  but  the  excess  of  this  above  that  due  to  the  head  must 
be  expended  in  overcoming  the  atmospheric  pressure  on  the 
outer  end  of  the  tube,  so  that  in  no  case  does  the  available  ex- 


I4O  FLOW   THROUGH   TUBES.  [CHAP.  VI. 

ceed  the  theoretic  energy.      No  contradiction  of  the  law  of 
conservation  therefore  exists. 

To  render  this  explanation  more  definite,  let  the  extreme 
case  be  considered  where  a  complete  vacuum  exists  near  the 
inner  end  of  the  adjutage,  if  that  were  possible,  as  it  perhaps 
might  be  with  a  tube  of  a  certain  form.  Let  h  be  the  head  of 
water  in  feet  on  the  centre  of  the  smallest  section.  The  mean 
atmospheric  pressure  on  the  water  in  the  reservoir  is  equivalent 
to  a  head  of  34  feet  (Art.  4).  Hence  the  total  head  which 
causes  the  discharge  into  the  vacuum  is  h  -f-  34  and  the 
velocity  of  flow  is  nearly  V2g(k  +  34).  Neglecting  the  re- 
sistances, which  are  very  slight  if  the  entrance  be  curved,  the 
coefficients  of  velocity  arid  discharge  can  now  be  found  ;  thus : 


For  h  =  loo,         v  =  V2g  X  134  =  1.16  ¥2gh  J 


For  h  =     10,         v  =  V2g  X    44  =  2.10  V2gh  ; 


For  k—  i,  v  =  V2g  X  35  =  5-92  ^2^/1. 
The  coefficient  hence  increases  as  the  head  decreases.  That 
this  is  not  the  case  in  the  above  experiments  is  undoubtedly 
due  to  the  fact  that  the  vacuum  was  only  partial,  and  that  the 
degree  of  rarefaction  varied  with  the  velocity.  The  cause  of  the 
vacuum,  in  fact,  is  to  be  attributed  to  the  velocity  of  the 
stream,  which  by  friction  removes  a  part  of  the  air  from  the 
inner  end  of  the  adjutage. 

It  follows  from  this  explanation  that  the  phenomena  of  in- 
creased discharge  from  a  compound  tube  could  not  be  pro- 
duced in  the  absence  of  air.  The  experiment  has  been  tried 
on  a  small  scale  under  the  receiver  of  an  air-pump,  and  it  was 
found  that  the  actual  flow  through  the  narrow  section  dimin- 
ished the  more  complete  the  rarefaction.  It  also  follows  that  it  is 
useless  to  state  any  value  as  representing,  even  approximately, 
the  coefficient  of  discharge  for  such  tubes.  To  secure  the  high- 
est coefficients,  it  is  thought  that  the  form  of  the  adjutage  of 


ART.  65.] 


INWARD  PROJECTING    TUBES. 


141 


FIG. 


43- 


the  compound  tube  should  not  be  conical,  but  of  the  shape  de- 
duced for  the  perfect  nozzle  in  Art.  63.  The  converging  part 
should  also  properly  be  of  the 
same  form.  Then  the  stream 
both  in  contracting  and  in  ex- 
panding follows  the  law  of 
the  perfect  jet ;  and  hence  it 
may  be  supposed  that  the  least  loss  of  energy  will  result,  and 
consequently  the  greatest  flow.  This,  however,  is  a  mere 
hypothesis,  not  yet  confirmed  by  experiment. 

Prob.  84.  Compute  the  pressure  per  square  inch  in  the 
section  BB  of  FRANCIS'  tube  when  h  =  1.36  feet  and  c  •=  2.43. 
What  is  the  height  of  the  column  CD  (Fig.  19,  Art.  27)  that 
could  be  lifted  by  a  small  pipe  inserted  at  BB? 

ARTICLE  65.  INWARD  PROJECTING  TUBES. 

Inward  projecting  tubes,  as  a  rule,  give  a  less  discharge 
than  those  whose  ends  are  flush  with  the  sides  of  the  reser- 
voir, due  to  the  greater  convergence  of  the  lines  of  direction 
of  the  filaments  of  water.  At  A  and  B  are  shown  inward  pro- 
jecting tubes  so  short  that  the  water  merely  touches  their  inner 
edges,  and  hence  they  may  more  properly  be  called  orifices. 
Experiment  shows  that  the  case  at  A,  where  the  sides  of  the 
tube  are  normal  to  the  side 
of  the  reservoir,  gives  the 
minimum  coefficient  of  dis- 
charge c  =  0.5,  while  for  B 
the  value  lies  between  0.5 
and  that  for  the  standard 
orifice  at  C.  The  inward 
projecting  cylindrical  tube 
at  D  has  been  found  to  give  FIG.  44. 

a  discharge  of  about  72  per  cent  of  the  theoretic  discharge, 
while  the  standard  tube  (Art.  61)  gives  82  per  cent.     For  the 


O— U.5U 


142  FLOW   THROUGH   TUBES.  [CHAP.  VI. 

tubes  E  and  F  the  coefficients  depend  upon  the  amount  of 
inward  projection,  and  they  are  much  larger  than  0.72  for 
both  cases,  when  computed  for  the  area  of  the  smaller  end. 

It  is  usually  more  convenient  to  allow  a  water-main  to  pro- 
ject inward  into  the  reservoir  than  to  arrange  it  with  its  mouth 
flush  to  a  vertical  side.  The.  case  D,  in  Fig.  44,  is  therefore  of 
practical  importance  in  considering  the  entrance  of  water  into- 
the  main.  As  the  end  of  such  a  main  has  a  flange,  forming  a 
partial  bell-shaped  mouth,  the  value  of  c  is  probably  higher 
than  0.72.  The  usual  value  taken  is  0.82,  or  the  same  as  for 
the  standard  tube  (Art.  61).  Practically,  as  will  be  seen  in  a 
later  article,  it  makes  little  difference  which  of  these  is  used, 
as  the  velocity  in  such  a  pipe  is  slow  and  the  resistance  at  the 
mouth  is  very  small  compared  with  the  frictional  resistances 
along  its  length. 

Prob.  85.  Find  the  coefficient  of  discharge  for  a  tube  whose 
diameter  is  one  inch,  when  the  flow  under  a  head  of  9  feet  is 
22.1  cubic  feet  in  3  minutes  and  30  seconds. 

ARTICLE  66.  EFFECTIVE  HEAD  AND  LOST  HEAD. 

The  terms  energy  and  head  are  often  used  as  equivalent 
although  really  energy  is  proportional  to  head.  Thus,  if  h  be 

the  head  on  an  orifice  or  tube,  —  the  velocity  head  of  the  issu- 
ing jet,  and  J^the  weight  of  water  discharged  per  second,  the 
theoretic  energy  per  second  is  Wh,  the  effective  or  actual  en- 
ergy is  W — ,  and  the  lost  energy  is  W  \h j .  It  is  more 

convenient  to  deal  directly  with  the  heads,  omitting  the  W\ 

v* 
thus  the  effective  head  in  this  case  is  — ,  and  the  lost  head  is 

2? 


ART.  66.]  EFFECTIVE  HEAD  AND  LOST  HEAD.  143 

If  no  losses  occur  due  to  friction,  contraction,  or  other 
causes,  the  effective  head  at  any  point  of  a  tube  or  pipe  is 
equal  to  the  hydrostatic  head  h.  This  effective  head  may  be 
exerted  either  in  producing  pressure  or  in  producing  velocity, 
or  part  of  it  in  pressure  and  part  in  velocity.  Thus,  as  shown 
in  Art.  27, 


where  hl  is  the  pressure-head  at  the  place  considered.  If 
there  be  no  motion  of  the  water  h  equals  /^  ,  and  if  the  flow  is 

so  rapid  that  there  be  no  pressure  h  equals  —  .     Owing  to  the 

o 

various  resistances,  however,  the  effective  head  /zx  -|  --  is  gen- 

erally less  than  the  total  head  //,  and  the  difference  is  called  the 
lost  head.  Thus,  at  any  section  of  a  tube  or  pipe  the  head 
which  has  been  lost  is 


At  the  end  of  the  tube,  or  rather  outside  of  the  tube,  there 
can  be  no  pressure  on  the  jet,  and  the  loss  of  head  in  the  flow 
of  the  jet  hence  is 

h'  =  h-—.  (41)' 

zg 

Thus  in  Art.  46  it  was  shown  that  for  the  standard  orifice 
the  loss  of  energy  or  head  is  about  4  per  cent,  and  in  Art.  61  it 
was  shown  that  for  the  standard  tube  the  loss  is  about  33 
per  cent. 

In  any  case  the  loss  of  head  in  a  jet  from  a  tube  or  orifice 
depends  merely  on  the  loss  of  velocity.  Let  cl  be  the  coeffi- 
cient of  velocity  :  then  for  a  small  orifice  or  tube 

V  =  Ci  V2gk, 


144  FLOW   THROUGH   TUBES.  [CHAP.  VI. 

and  the  effective  velocity-head  is 
Consequently  the  loss  of  head  is 

o^r          ^  i  /     *  \~r    / 

It  is  sometimes  more  convenient  especially  for  pipes  to  express 
this  loss  in  terms  of  the  velocity-head.  The  value  of  h  in  terms 
of  this  is 

4=i~, 

and  hence  the  loss  of  head  is 


in  which  v  is  the  actual  velocity  of  discharge. 

For  the  standard  tube  (Fig.  38,  Art.  61)  the  coefficient  of 
velocity  is  equal  to  the  coefficient  of  discharge  whose  mean 
value  is  0.82.  The  effective  head  of  the  jet  then  is 

—  =  (o.82)V/  =  0.67/2, 

e> 

and  the  loss  of  head  is 

h'  =  (i  —  o.6;)/z      =  0.33^, 
or 

,.       /    i  \  v*  v* 

h'  =  ( — i )  —  =  0.49  — . 

\o.6;        )  2g  *  2g 

Hence  the  loss  of  head  may  be  said  to  be  either  33  per  cent 
of  the  total  head  or  49  per  cent  of  the  effective  velocity-head  ; 
that  is,  the  lost  energy  is  about  one-third  of  the  total  energy 
or  about  one-half  of  the  effective  energy. 


ART.  67.]  LOSSES  IN   THE   STANDARD    TUBE.  '    145 

In  reality,  work  or  energy  is  never  lost,  but  is  merely  trans- 
formed into  other  forms  of  energy.  In  the  tube  the  one-third 
of  the  total  energy  which  has  been  called  lost  is  only  lost 
because  it  cannot  be  utilized  as  work ;  it  is,  in  fact,  transformed 
into  heat,  which  raises  the  temperature  of  the  water.  And  so 
it  is  in  all  cases  of  lost  head :  the  pressure-head  plus  the 
velocity-head  is  the  effective  head  which  can  alone  be  rendered 
useful ;  if  this  be  less  than  the  total  hydrostatic  head,  the 
remainder  has  disappeared  in  heat. 

Prob.  86.  Show  that  the  lost  head  is  nearly  equal  to  the 
effective  head  for  an  inward  projecting  cylindrical  tube. 

ARTICLE  67.  LOSSES  IN  THE  STANDARD  TUBE. 

The  loss  of  head  in  the  flow  from  the  short  cylindrical  tube 
is  large,  but  not  so  large  as  might  be  expected  from  theoretical 
considerations  based  on  the  known  coefficients  for  orifices. 
If  the  tube  has  a  length  of  only  two  diameters  the  jet  does 
not  touch  its  inner  surface,  and  the  flow  occurs  as  from  a 
standard  orifice.  The  velocity  in  the  plane  of  the  inner  end 
is  then  61  per  cent  of  the  theoretic  velocity,  since  the  mean 
coefficient  of  discharge  is  0.61.  Now  if  the  tube  be  increased 
in  length  about  one  diameter  its  outer  end  is  filled  by  the  jet, 
and  since  the  contraction  still  exists,  it  might  be  inferred  that 
the  coefficient  for  that  end  would  be  also  0.61  :  this  would 
give  an  effective  head  of  (0.6 i)*h  or  O.37//,  so  that  the  loss  of 
head  would  be  0.63^.  Actually,  however,  the  coefficient  is 
found  to  be  0.82  and  the  loss  of  head  only  0.33^.  It  hence 
appears  that  further  explanation  is  needed  to  account  for  the 
increased  discharge  and  energy. 

It  is  to  be  presumed,  in  the  first  place,  that  a  loss  of  about 
0.04/1  occurs  at  the  inner  end  of  the  tube  in  the  same  manner 
as  in  the  standard  orifice,  due  to  retardation  of  the  outer  fila- 
ments (Art.  46).  The  effective  head  at  the  contracted  section 


146 


FLOW    THROUGH    TUBES. 


[CHAP.  VI. 


in  the  tube  is  then  about  0.96/2.  If  the  coefficient  of  contrac- 
tion have  the  value  0.62,  as  in  the  orifice,  the  velocity  in  that 
section  is  greater  than  at  the  end  of  the  tube,  and,  since  the 
velocities  are  inversely  as  the  areas  of  the  sections,  that  velo- 
city is 

Vl  =  0^62  ^2gk  =  I'32  l/^' 

which  is  nearly  one-third  larger  than  the  theoretic  velocity. 
The  velocity-head  at  that  section  then  is 

3l  = 

2*r " 

and  consequently  the  pressure-head  is      . 

/*,  —  0.96^  —  1.75^  =  —  0.79^. 

There  exists  therefore  a  negative  pressure  or  partial  vacuum 
in  the  tube  which  is  sufficient  to  lift  a  column  of  water  to  a 
height  of  about  three-fourths  the  head. 
This  conclusion  has  been  confirmed  by 
experiment  for  low  heads,  and  was  in 
fact  first  discovered  experimentally  by 
VENTURI.  For  high  heads  it  is  not 
valid,  since  in  no  event  can  atmospheric 
pressure  raise  a  column  of  water  higher 
than  about  34  feet  (Art.  4) ;  probably 
under  high  heads  the  coefficient  of  con- 
traction of  the  jet  in  the  tube  becomes 
much  greater  than  0.62. 


FIG.  45- 


The  reason  of  the  increased  discharge  of  the  tube  over  the 
orifice  is  hence  due  to  the  negative  pressure  or  partial  vacuum, 
which  causes  a  portion  of  the  atmospheric  head  of  34  feet  to 
be  added  to  the  head  h,  so  that  the  flow  at  the  contracted 
section  occurs  as  if  under  the  head  h  -j-  7z, ,  as  in  the  diverging 


ART.  67.]  LOSSES  IN   THE   STANDARD    TUBE.  Itf 

tube  (Art.  64).  The  occurrence  of  the  partial  vacuum  is  attrib- 
uted to  the  friction  of  the  sides  of  the  jet  on  the  air.  When 
the  flow  begins,  the  jet  is  surrounded  by  air  of  the  normal 
atmospheric  pressure  which  is  imprisoned  as  the  jet  fills  the 
tube.  The  friction  of  the  moving  water  carries  some  of  this 
air  out  with  it,  thus  rarefying  the  remaining  air.  This  rarefac- 
tion, or  negative  pressure,  is  followed  by  an  increased  velocity 
of  flow,  and  the  process  continues  until  the  air  around  the  con- 
tracted section  is  so  rarefied  that  no  more  is  removed,  and  the 
flow  then  remains  permanent,  giving  the  results  ascertained  by 
experiment.  The  experiments  of  BUFF  have  proved  that  in 
an  almost  complete  vacuum  the  discharge  of  the  tube  is  but 
little  greater  than  that  of  the  orifice.* 

The  velocity-head  in  the  contracted  section  of  the  jet  is  thus 
about  1.75/2,  but  of  this  0.79/2  must  be  expended  in  overcom- 
ing the  atmospheric  pressure  at  the  end  of  the  tube,  so  that 
the  effective  head  is  only  0.96/2.  If  the  retarding  influence  of 
the  outer  end  be  0.04/2,  or  the  same  as  that  of  the  inner  end, 
the  effective  head  is  reduced  to  0.92/2,  while  the  actual  effect- 
ive velocity-head  is  0.67/2.  Thus  a  further  loss  of  0.25/2  is  to 
be  accounted  for,  and  this  must  be  supposed  to  be  due  to  the 
enlargement  of  the  section  of  the  jet,  and  the  consequent  dimi- 
nution of  velocity,  whereby  the  energy  is  converted  into  heat. 
The  partial  vacuum  causes  neither  a  gain  nor  loss  of  head,  and 
the  only  losses  are  0.04/2  at  the  inner  end  of  the  tube,  0.25/2 
in  the  enlargement  of  the  jet,  and  0.04/2  at  the  outer  end,  or 
in  all  0.33/2.  These  quantities,  of  course,  are  only  approxi- 
mate, as  they  depend  upon  the  mean  coefficients  0.98,  0.62, 
and  0.82,  all  of  which  are  liable  to  variation. 

Prob.  87.  Discuss  the  losses  of  head  in  an  inward  projecting 
tube,  taking  c'  =  0.6  and  c  =  0.7. 

*  See  RUHLMANN'S  Hydromechanik  (Hannover,  1879). 


148 


FLOW   THROUGH    TUBES. 


[CHAP  VI. 


ARTICLE  68.  Loss  DUE  TO  ENLARGEMENT  OF  SECTION. 

When  a  tube  or  pipe  is  kept  constantly  full  of  water  a  loss 
of  head  is  found  to  result  when  the  section  is  enlarged  so  that 

the  velocity  is  diminished.  Let 
vl  and  v^  be  the  velocities  in  the 
smaller  and  larger  sections,  arid 
h^  and  //2  the  corresponding  pres- 
sure-heads. The  effective  head 
in  the  first  section  is  the  sum  of 
the  pressure-  and  velocity-heads 
(Arts.  2J  and  66),  or 

FIG.  46.  #!  -h 

and  the  effective  head  in  the  second  section  is 


If  no  losses  occur,  these  two  expressions  are  equal ;  but  as  the 
second  effective  head  is  always  smaller  than  the  first,  their  dif- 
ference is  the  loss  of  head  between  the  two  sections,  or  the 
lost  head  h1  is 


h'  = 


•    •    •    •    (43) 


This  is  a  general  expression,  which  gives  the  loss  of  head  due 
not  only  to  enlargement,  but  to  all  resistances  between  any  two 
sections  of  a  horizontal  tube  or  pipe.  If  the  difference  /za  —  kl 
of  the  pressure  columns  shown  in  Fig.  46  is  measured,  and  the 
velocities  determined,  the  loss  of  head  is  thus  found  in  any 
particular  case. 


The  loss  of  head  due  to  the  sudden  enlargement  of  section, 


ART.  68.]  LOSS  DUE    TO  ENLARGEMENT  OF  SECTION. 


149 


or  rather  to  the  sudden  diminution  of  velocity  caused  by  the 

enlargement,  can  be  expressed  by  the  for-  M 

mula 


To  prove  this,  let/!  be  the  unit  pressure 
in  AB  and  /3  that  in  CD.  At  a  section 
MN  very  near  the  place  of  enlargement 
the  unit  pressure  is  also  pl  ,  since  the  velocity  v^  is  maintained 
for  a  short  distance  after  leaving  AB,  its  direction,  however, 
being  changed  so  as  to  form  eddies.  Let  ay  be  the  area  of 
the  section  CD  or  MN.  Then  the  pressure  which  acts  in  the 
opposite  direction  to  the  flow  is  a9(j>9  —  A)>  anc*  this  ls  tne 
force  which  causes  the  velocity  to  diminish  from  vl  to  vz.  Now 
in  Art.  32  it  was  shown  that  the  force  which  causes  W  pounds 

Wv 
of  water  to  increase  in  velocity  from  o  to  v  is  --  ,  and  con- 

versely the  same  force  applied  in  the  opposite  direction  will 
cause  the  velocity  to  diminish  from  v  to  o.  Therefore  the 
value  of  the  pressure  ajj>9  —  /,)  is 


«,(A  -  A)  =  -      -  -,)  = 

<5  o 

where  w  is  the  weight  of  a  cubic  unit  of  water.     This  expres- 
sion may  be  written, 

A  _  A  _  y.(yi  -  y«) 


or  (Art.  9) 

This  value  of 
reduces  it  to 


—  /^  inserted   in  the  general  equation   (43) 


(44) 


ISO  FLOW  THROUGH   TUBES-.  [CHAP.  VI. 

which  is  the  formula  for  loss  of  head  due  to  sudden  enlarge- 
ment. The  loss  of  energy  in  this  case  is  similar  to  that  which 
occurs  in  the  impact  of  inelastic  bodies,  work  being  converted 
into  heat. 


Let  al  and  #3  be  the  areas  of  the  cross-sections  AB  and  CD. 

en  vv  =  — 2-^2 ,  an< 
a\ 

largement  becomes 


Then  vl  =  — 2-^2 ,  and  the  formula  for  loss  of  head  in  sudden  en- 
a\ 


which  is  often  a  more  convenient  form  for  practical  use.     If 
al  =  #2  or  if  v9  =  o  no  loss  of  head  results. 

If  a  gradual  enlargement  of  section  be  made  so  that  no  im- 
pact occurs,  the  energy  due  to  the  velocity  vl  is  slowly  changed 
into  pressure,  so  that  head  is  not  lost.  There  is,  however,  no 
distinct  line  of  division  between  sudden  and  gradual  enlarge- 
ment, and  for  a  case  like  Fig.  46  experiment  can  alone  deter- 
mine the  value  of  hl  —  7*2  and  the  loss  of  head.  In  the  last 
article  it  was  seen  that  about  0.2 ^h  is  lost  in  the  expansion  of 
the  jet  between  the  contracted  section  and  the  end  of  the  tube. 
This  seems  like  a  case  of  gradual  enlargement,  but  as  no  pres- 
sure can  exist  at  the  end  of  the  tube  the  loss  of  head  must 
be  the  same  as  for  sudden  enlargement  of  section ;  in  fact 
^  =  1.32  V2gh  and  v^  =  0.82  V2gh,  whence  by  the  above 
formula  h'  =  o.2$h. 

The  loss  of  head  due  to  sudden  enlargement  may  often  be 
very  great,  as  the  following  example  will  show.  Let  the  effec- 
tive head  in  the  section  AB  be  h,  all  of  which  exists  as  velocity, 
so  that  v^  =  V2gh ;  let  the  diameter  of  AB  be  2  inches,  and 
that  of  CD  be  4  inches,  so  that  the  area  at  CD  is  four  times 


ART.  69.]     LOSS  DUE    TO   CONTRACTION  OF  SECTION. 


that  at  AB,  and  hence  the  velocity  in  CD  is  v^  =  J-  V~2gh.     The 
loss  of  head  then  is 


so  that  more  than  half  the  energy  of  the  water  in  AB  is  lost  in 
shock  or  impact.  At  CD  the  effective  head  is  then  ^//,  of 
which  y1^  is  velocity-head  and  -f$h  is  pressure-head.  Sudden 
enlargement  of  section  is  therefore  to  be  avoided. 

Prob.  88.  In  a  horizontal  tube  like  Fig.  46  the  diameters  are 
6  inches  and  12  inches,  and  the  heights  of  the  pressure-columns 
or  piezometers  are  '12.16  feet  and  12.96  feet  above  the  same 
bench  mark.  Find  the  loss  of  head  between  the  two  sections 
when  the  discharge  is  1.57  cubic  feet  per  second,  and  also  when 
it  is  4.71  cubic  feet  per  second. 

ARTICLE  69.  Loss  DUE  TO  CONTRACTION  OF  SECTION. 

When  a  sudden  contraction  of  section  in  the  direction  of 
the  flow  occurs,  as  in  Fig.  48,  the  water  suffers  a  contraction 
similar  to  that  in  the  standard  tube,  and  hence  in  its  expansion 
to  fill  the  smaller  section  a  loss  of  head 
results.  Let  vl  be  the  velocity  in  the 
larger  section  and  v  that  in  the  smaller, 
while  v'  is  the  velocity  in  the  contracted 
section  of  the  flowing  stream ;  and  let  a: , 
a,  and  a'  be  the  corresponding  areas  of 
the  cross-sections.  From  the  formula 
(44)'  of  the  last  article  the  loss  of  head 
due  to  the  expansion  of  section  from  a' 
to  a  is  FlG-48. 


57=?-'    2T;     '    '    '    (45> 


in  which  c1  is  the  coefficient  of  contraction  or  the  ratio  of  a' 
to  a. 


152 


FLOW   THROUGH    TUBES. 


[CHAP.  VI. 


The  value  of  cr  depends  upon  the  ratio  between  the  areas 
a  and  ar  When  a  is  small  compared  with  #,,  the  value  of  c' 
may  be  taken  at  0.62  as  for  orifices  (Art.  35).  When  a  is 
equal  to  a1  there  is  no  contraction  or  expansion  of  the  stream, 
and  c'  is  unity.  Let  d  and  dl  be  the  diameters  corresponding 
to  the  areas  a  and  al ,  and  let  r  be  the  ratio  of  d  to  d^ .  Then 
experiments  seem  to  indicate  that  an  expression  of  the  form 


1    I.I  —  r 

gives  the  law  of  variation  of  c'  with  r.  Determining  the  values 
of  ;;/  and  n  from  the  two  limiting  conditions  above  stated, 
there  is  found, 

0.0418 

cf  —  0.582  H —  - — , 

i.i  —  r* 

from  which  approximate  values  of  c'  can  be  computed.  The 
manner  of  the  variation  in  the  values  of  c'  is  indicated  by  the 
following  tabulation  : 

For  r  =  o.o,      0.2,      0.4,      0.6,      0.7,      0.8,      0.9,       i.o, 
c'  =  0.62,    0.63,    0.64,    0.67,    0.69,    0.72,    0.79,     i.oo. 

from  which  intermediate  values  may  often  be  taken  without 
the  necessity  of  using  the  formula. 

For  a  case  of  gradual  contraction  of  section,  such  as  shown 
in  Fig.  49,  the  loss  of  head  is  less  than  that  given  by  the  above 

formula,  and  can  only  be  de- 
termined for  a  given  velocity  of 
flow  by  observing  the  difference 
of  the  heights  of  the  pressure 
columns. 


The  loss  of  head  then 


is 


FIG.  49. 


ART.  70.]  PIEZOMETERS.  153 

as  proved  in  Art.  68.     This  may  be  written 


If  the  change  of  section  be  made  so  that  the  stream  has  no 
subsequent  enlargement,  loss  of  head  is  avoided,  for,  as  the 
above  discussions  show,  it  is  the  loss  in  velocity  due  to  sudden 
expansion  which  causes  the  loss  of  head. 

The  loss  due  to  sudden  contraction  of  a  tube  or  pipe  is 
usually  much  smaller  than  that  due  to  sudden  enlarge- 
ment. For  instance,  if  the  diameter  of  the  larger  section 
be  three  times  that  of  the  smaller,  and  the  velocity  in  the 
large  section  be  2  feet  per  second,  the  loss  of  head  when  the 
flow  passes  from  the  smaller  to  the  larger  section  is 


=  .0  fee, 


But  if  the  flow  takes  place  in  the  opposite  direction  the  co- 
efficient c'  is  about  0.64,  and  the  loss  of  head  is 


=  (    '          tm  =  ,.6  feet, 
\o.6          /    2 


which  may  be  made  to  vanish  by  rounding  the  edges  where 
the  change  of  section  occurs. 

Prob.  89.  Compute  the  loss  of  head  when  a  pipe  which  dis- 
charges 1.57  cubic  feet  per  second  suddenly  diminishes  in  sec- 
tion from  12  to  6  inches  diameter. 

ARTICLE  70.  PIEZOMETERS. 

A  piezometer  is  an  instrument  for  measuring  the  pressure 
which  exists  in  a  pipe.  In  its  simplest  form  it  consists  merely 


154 


FLOW   THROUGH   TUBES. 


[CHAP.  VI. 


FIG.  49. 


of  a  glass  tube,  as  at  A,  in  which  the  water  rises  to  a  height  //,. 

At  B  is  a  form  where  the  tube 
connecting  with  the  pipe  is  of 
metal,  which  is  joined  by  a  flexi- 
ble hose  with  a  glass  tube,  which 
may  be  placed  alongside  of  a 
graduated  rod  to  read  the  height 
hl .  At  C  is  a  common  pressure 
gauge  whose  dial  is  graduated  so 
as  to  read  either  heights  or  pressures,  as  may  be  desired.  When 
//!  is  found  by  measurement,  the  pressure  per  square  unit  is 
computed  from  the  relation  pl  =  wh^  (Art.  9).  In  order  to 
secure  accurate  results  with  piezometers,  it  is  necessary  that 
they  be  inserted  into  the  pipe  exactly  at  right  angles ;  if  in- 
clined with  or  against  the  current,  the  height  //,  is  greater  or 
less  than  that  due  to  the  actual  pressure  at  the  mouth. 

If  no  loss  of  head  occurs  between  the  reservoir  and  the 
place  where  the  piezometer  is  inserted  the  velocity  and  dis- 
charge through  the  pipe  may  be  determined.  The  flow  being 
stopped,  the  water  in  the  piezometer  rises  to  the  height  /^  at 
the  same  level  as  the  surface  level  of  the  reservoir ;  when  the 
flow  occurs  it  stands  at  the  height  /2a.  Then 


whence 

v  —  V2g(hi  —  ^2),    ......     (46) 

and  hence  the  discharge  is  known  for  a  pipe  of  given  size.  It 
is  only  in  cases  of  low  velocities,  however,  that  this  method  of 
gauging  the  flow  is  at  all  applicable,  owing  to  the  losses  of 
head  which  always  exist. 

The  question  as  to  the  point  from  which  the  pressure-head 
should  be  measured  deserves  consideration.  In  the  figures  of 
the  preceding  articles  hl  and  //2  have  been  estimated  upward 


ART.  70.] 


PIEZOMETERS. 


155 


from  the  centre  of  the  tube,  and  it  is  now  to  be  shown  that 
this  is  probably  correct.  Let  Fig.  50  represent  a  cross-section 
of  a  tube  to  which  are  attached  three  piezome- 
ters as  shown.  If  there  be  no  velocity  in  the 
tube  or  pipe,  the  water  surface  stands  at  the  '/ZTZLT 


same  level  in  each  piezometer,  and  the  mean 
pressure-head  is  certainly  the  distance  of  that 
level  above  the  centre  of  the  cross-section.  If 
the  water  in  the  pipe  be  in  motion,  probably  FlG'  5°* 

the  same  would  hold  true.  Referring  to  formula  (43)  of  Art. 
68,  and  to  Fig.  46,  it  is  also  seen  that  if  there  be  no  velocity 
h'  —  kl  —  //2 ,  which  cannot  be  true  unless  kl  —  /z2  =  o,  since 
there  can  be  no  loss  of  head  in  the  transmission  of  static  pres- 
sures; hence  hl  and  h^  cannot  be  measured  from  the  top  of  the 
section.  In  any  event,  since  the  piezometer  heights  represent 
the  mean  pressures,  it  appears  that  they  should  be  reckoned 
upward  from  the  centre  of  the  section.  The  absolute  values 
of  h^  and  h^  are  not  generally  required,  the  difference  h^  —  7za 
being  alone  used  in  computations ;  nevertheless  the  above  con- 
siderations are.  not  unimportant. 

The  principal  application  of  the  piezometer  is  to  the  meas- 
urement of  losses  of.  head,  as  indicated  in  Art.  68  for  the  case 
of  horizontal  pipes.  The  same  method  applies  to  inclined 

pipes,  only  here  the  piezom-     A_    _    „ _B 

eter  readings  are  usually 
taken  above  an  assumed 
datum  MN,  as  shown  in 
Fig.  51.  Let  al  and  #2  be 
the  areas  of  any  two  sec- 
tions of  a  pipe,  vl  and  v3  the 
velocities,  H^  and  H^  the 

heights  of  the  piezometers      ^_ji A.- N 

above  a  datum  MN,  and  hv  FIG.  5i. 

and  7/2  the  heights  above  the  axis  of  the  pipes,  that  is,  the  mean 


156  FLOW   THROUGH   TUBES.  [CHAP.  VL 

pressure-heads.  When  no  flow  occurs  the  piezometers  stand 
in  the  same  level  line  AB.  When  the  flow  takes  place,  deliv- 
ering W  pounds  of  water  per  second,  the  effective  energy  in 
the  first  section  is 


and  that  in  the  second  section  is 

wl^  +  ^}. 


Now  let  2  be  the  vertical  distance  of  the  centre  of  the  second 
section  below  the  first.  Were  it  not  for  losses  the  energy  in 
the  second  section  would  be 


Therefore  the  energy  lost  in  heat  due  to  friction,  enlargement, 
contraction,  and  all  other  causes,  between  the  two  sections,  is 


or  the  loss  of  head  is 


*  2g         +*>+»-*•• 

But  from  the  figure  it  is  seen  that 

At  +  »-A,  =  ffl-ff,. 

Hence  the  loss  of  head  between  the  two  sections  is 


(47) 


or  the  same  as  shown  in  Art.  68  for  horizontal  tubes,  the  pie- 
zometer elevations  being  referred  to  the  same  datum. 

If  the  pipe  be  of  the  same  diameter  at  the  two  sections  the 
velocities  vl  and  v^  are  equal,  and  the  loss  of  head  is 

k'  =  Hl-H,,    ......    (47)' 

which  is  merely  the  difference  of  level  of  the  water  surfaces 


ART.  70.] 


PIEZOMETERS. 


157 


in  the  piezometers.  If  the  two  sections  are  at  the  same  eleva- 
tion,  or  if  the  second  section  is  lower  than  the  first,  this  loss  is 
entirely  due  to  resistances  which  convert  the  energy  into  heat. 
When,  however,  the  second  section  is  higher  than  the  first  by 
the  distance  #',  the  head  z'  is  lost  in  overcoming  the  force  of 
gravity,  and  the  remainder  h' — z'  is  the  portion  lost  in  heat. 
Piezometers  therefore  furnish  a  very  convenient  method  of  de- 
termining lost  head  in  pipes  of  uniform  section.  For  pipes  of 
varying  section  they  are  rarely  applied,  as  the  discharge  per 
second  must  be  measured  to  find  the  velocities  vl  and  vt . 

In  practice  it  is  usually  the  case  that  the  piezometric  tube 
is  simply  tapped  into  the  top  of  the  pipe  whose  flow  is  to  be 
investigated.  It  is  thought,  however,  that  this  may  not  give 
the  mean  pressure  throughout  the  section.  In  the  equations 
above  deduced  z\  and  vt  are  the  mean  velocities  in  the  two 
sections  and  /^  and  7z2  the  corresponding  mean  pressure-heads. 
In  order  that  the  piezometer  may  correctly  indicate  these 
mean  pressure-heads,  they  should  perhaps  be  connected  with 
the  pipe  at  the  sides  and  bottom  as  well  as  at  the  top.  Pie- 
zometric measurements  are  hence  liable  to  give  results  more  or 
less  uncertain. 

If  a  tube  be  inserted  obliquely  to  the  direction  of  the  cur- 
rent it  no  longer  indicates  the  true  pressure-head,  for  it  is 
found  that  the  height  of  the  water  is  greater  when  the  mouth 
of  the  tube  is  inclined  toward  the  current  than  when  inclined 
away  from  it.  Let  0 
be  the  angle  between 
the  direction  of  the 
flow  and  the  inserted 
tube.  Then  the  dy- 
namic pressure  in  the 
direction  of  the  flow 


FIG.  52. 


is  proportional   to   the  velocity-head,  and   the  component  of 


158  FLOW   THROUGH   TUBES.  [CHAP.  VL 

this  in  the  direction  of  the  tube  tends  to  increase  the  normal 
pressure-height  /^  when  6  is  less  than  90°  and  to  decrease  it 
when  0  is  greater  than  90°.  Thus 

*.  =  *,+  J  cos  e 

may  be  written  as  approximately  applicable  to  the  two  cases. 
In  this,  if  the  tube  be  inserted  normal  to  the  pipe,  0  =  90°  and 
/20  becomes  /^  ,  the  height  due  to  the  static  pressure  in  the 
pipe ;  if  v  =  o,  the  angle  6  has  no  effect  upon  the  piezometer 
readings.  This  discussion  indicates  that  when  the  velocity  v 
is  great,  piezometric  measurements  may  be  affected  with  errors 
if  the  connection  be  not  made  truly  normal  to  the  direction  of 
the  flow. 

Prob.  90.  In  one  of  the  experiments  on  the  compound  tube 
shown  in  Fig.  53  the  areas  of  the  sections  al  and  as  were 
57.823  square  feet,  while  that  of  #a  was  7.047  square  feet. 
When  the  discharge  was  54.02  cubic  feet  per  second  the  pie- 
zometric elevations  were : 

H,  =  99.838,         H,  =  98.921,         ffs  =  99.736  feet. 

Show  that  the  head  lost  was  0.017  feet  between  al  and  at,  and 
0.085  feet  between  #a  and  as . 

ARTICLE  71.  THE  VENTURT  WATER  METER. 

It  has  been  shown  by  HERSCHEL*  that  a  compound  tube 
provided  with  piezometers  may  be  used  for  the  accurate 
measurement  of  water.  The  apparatus,  which  is  called  by  him 
the  VENTURI  Water  Meter,  is  shown  in  outline  in  Fig.  53,  and 
consists  of  a  compound  tube  (Art.  64)  terminated  by  cylinders, 
into  the  top  of  which  are  tapped  the  piezometers  H^  and  Hy 
Surrounding  the  small  section  az  is  a  chamber  into  which  four 

*  Transactions  American  Society  of  Civil  Engineers,  1887,  vol.  xviii.  p.  228. 


ART.  71.] 


THE    VENTURI    WATER  METER. 


159 


or  more  holes  lead  from  the  top,  bottom,  and  sides  of  the  tube, 
and  from  which  rises  the  piezometer  H^  .  The  flow  passing 
through  the  tube  has  the  velocities  vltvt,  and  v^  at  the  sections 
and  az  ,  and  these  velocities  are  inversely  as  the  areas  of 


a, 


the  sections  (Art.  19).     When  the  pressure  in  at  is  positive  the 


FIG.  53. 

water  stands  in  the  central  piezometer  at  a  height  //, ,  as  shown 
in  the  figure ;  when  the  pressure  is  negative  the  air  is  rarefied, 
and  a  column  of  water  lifted  to  the  height  h^ .  If  E  is  the 
height  of  the  top  of  the  section  #2  above  the  datum  the  value 
of  H9  for  the  case  of  negative  pressure  was  taken  to  be  E  —  h^ . 
The  apparatus  was  constructed  so  that  the  areas  #,  and  aa  were 
equal,  while  02  was  about  one-ninth  of  these. 

To  determine  the  discharge  per  second  through  the  tube, 
the  areas  at  and  #2  are  to  be  accurately  found  by  measure- 
ments of  the  diameters  ;  then 

Q  =  a^  ,     or     Q  =  a^ . 

If  no  losses  of  head  occur  between  the  sections  al  and  a^  the 
quantity  ti  in  the  formula  of  the  last  article  is  o,  and 


o  = 


l6o  FLOW   THROUGH   TUBES.  [CHAP.  VI. 

Inserting  in  this  for  ^  and  v^  their  values  in  terms  of  Q,  and 
then  solving  for  Q,  gives  the  result 


which  may  be  called  the  theoretic  discharge.  Dividing  this 
expression  by  al  gives  the  velocity  vlt  and  dividing  it  by  az 
gives  the  velocity  ^2 .  Owing  to  the  losses  of  head  which 
actually  exist,  this  expression  is  to  be  multiplied  by  a  coeffi- 
cient c\  thus: 


•^-f^2^.-^)   •  •  •  (48) 

is  the  formula  for  the  actual  discharge  per  second. 

Reference  is  made  to  HERSCHEL'S  paper,  above  quoted,  for 
a  full  description  of  the  method  of  conducting  the  experi- 
ments. The  discharge  was  actually  measured  either  in  a  large 
tank  or  by  a  weir  ;  and  thus  q  being  known  for  observed  pie- 
zometer heights  //!  and  H^  ,  the  value  of  c  was  computed  by 
dividing  the  actual  by  the  theoretic  discharge.  For  example, 
the  smaller  tube  used  had  the  areas- 

al  =  0.77288,         0a  =  0.08634  square  feet  ; 
hence  the  theoretic  discharge  is 

Q  =  0.086884  \/2g(H,  -  //,), 
and  the  coefficient  of  discharge  or  velocity  is 


In  experiment  No.  I  the  value  of  Hl  was  99.069,  while  7z2  was 
24.509  feet,  and  the  actual  discharge  was  4.29  cubic  feet  per 
second.  As  E  was  84.704,  the  value  of  //"„  is  60.195  feet.  The 
theoretic  discharge  then  is 

Q  =  0.086884  X  8.02  1/38.874  =  4.345. 


ART.  71.]  THE    VENTURI    WATER  METER.  l6l 

Dividing  4.29  by  this,  gives  for  c  the  value  0.988.  Fifty-five 
experiments  made  in  this  manner,  in  all  of  which  negative 
pressure  existed  in  #3 ,  gave  coefficients  ranging  in  value  from 
0.94  to  1.04,  only  four  being  greater  than  i.oi  and  only  two 
less  than  0.96. 

The  larger  tube  used  had  the  areas  al  =  57.823  and 
a^  =  7.074  square  feet,  and  the  pressure  at  the  central  piezom- 
eter was  both  positive  and  negative.  Twenty-eight  experi- 
ments give  coefficients  ranging  from  0.95  to  0.99,  the  highest 
coefficients  being  for  the  lowest  velocities.  In  this  tube  the 
velocity  at  the  section  #Q  ranged  from  5  to  34.5  feet  per  second. 
The  small  variation  in  the  coefficients  for  the  large  range  in 
velocity  indicates  that  the  apparatus  may  in  the  future  take  a 
high  rank  as  an  accurate  instrument  for  the  measurement  of 
water.  Under  low  velocities,  however,  it  is  not  probable  that 
the  arrangement  of  piezometers  shown  in  Fig.  53  will  give  the 
best  results ;  in  order  that  H^  may  correctly  indicate  the  mean 
pressure  in  al ,  connection  seems  to  be  required  both  at  the 
bottom  and  sides  of  the  tube  like  that  at  #a .  It  is  thought, 
moreover,  that  the  elevation  E  should  be  measured  to  the 
centre  of  the  section  rather  than  to  the  top.  The  lower  pie- 
zometer HI  is  not  an  essential  part  of  the  apparatus  and  may 
be  omitted,  although  it  was  of  value  in  the  experiments  as  show- 
ing the  total  loss  of  head. 

Prob.  91.  Given  a^  —  7.074  and  al  =  57.823  square  feet, 
hn_  —  12.204,  E  =  90.909,  and  //",  =  98.773  feet,  to  compute 
the  coefficient  of  discharge  when  q  =  243.87  cubic  feet. 


162 


FLOW  THROUGH  PIPES. 


[CHAP.  Mil 


CHAPTER  VII. 
FLOW  THROUGH  PIPES. 

ARTICLE  72.  FUNDAMENTAL  IDEAS. 

The  simplest  case  of  flow  through  a  pipe  is  that  where  the 
discharge  occurs  entirely  at  the  end,  there  filling  the  entire  sec- 
tion, as  in  a  tube ;  such  pipes  are  said  to  be  in  a  condition  of 
full  flow.  Other  cases  are  those  where  the  discharge  is  drawn 
from  the  pipe  at  several  points  along  its  length,  as  in  the  water 
mains  for  the  supply  of  towns.  Pipes  with  full  flow  will  be 
first  considered,  but  most  of  the  principles  and  tables  relating 
to  them  apply  with  but  slight  modification  to  water  mains. 
Pipes  used  in  engineering  practice  are  rarely  less  than  f  inch 
in  interior  diameter,  and  may  range  from  this  value  upward  to 
4  feet  or  more. 

The  phenomena  in  a  pipe  with  full  flow  are  apparently  sim- 
ple. The  water  from  the  reservoir,  as  it  enters  the  pipe,  suffers 

more  or  less  contraction,  depend- 
ing upon  the  manner  of  connec- 
tion, as  in  tubes.  Its  velocity  is 
then  retarded  by  the  resistances 
of  friction  and  cohesion  along  the 
interior  surface,  so  that  the  dis- 
charge at  the  end  is  much  smaller 

than  in  the  tube.  When  the  flow  becomes  permanent  the  pipe 
is  entirely  filled  throughout  its  length ;  and  hence  the  mean 
velocity  at  any  section  is  the  same  as  that  at  the  end,  if  the 
size  be  uniform.  This  velocity  is  found  to  decrease  as  the 


ART.  72.]  FUNDAMENTAL  IDEAS.  163 

length  of  the  pipe  increases,  other  things  being  equal,  and  be- 
comes very  small  for  great  lengths,  which  shows  that  nearly  all 
the  head  has  been  lost  in  overcoming  the  resistances. 

The  head  which  causes  the  flow  is  the  difference  in  level 
from  the  surface  of  the  water  in  the  reservoir  to  the  centre  of 
the  end,  when  the  discharge  occurs  freely  into  the  air.  If  h  be 
this  head,  and  W  the  weight  of  water  discharged  per  second, 
the  theoretic  energy  per  second  is  Wh  ;  and  if  v  be  the  actual 


velocity  of  discharge  the  effective  energy  is  -  .     The  lost 


energy  is  then  Wyi  —  —  ),  and  this  has  disappeared  in  heat  in 
overcoming  the  resistances.  In  other  words,  the  total  head  is 
h,  the  effective  head  of  the  outflowing  stream  is  —  ,  and  the 

lost  head  is  h  ---  .     If  the  lower  end  of  the  pipe  is  sub- 

Zg 

merged,  as  is  often  the  case,  the  head  h  is  the  difference  in 
elevation  between  the  two  water  levels. 

The  length  of  a  pipe  is  measured  along  its  axis,  following  all 
its  windings  if  any.  When  the  length  is  about  two  and  one- 
half  diameters  the  pipe  is  a  tube  whose  coefficient  of  discharge 
varies  from  0.71  to  0.82,  according  to  the  arrangement  of  its 
inner  end  (Art.  65).  As  the  length  increases  the  coefficient  of 
discharge  becomes  less  than  from  the  tube,  and  for  long  pipes 
it  becomes  very  small  indeed  —  indicating  that  the  greater  part 
of  the  head  h  is  expended  in  heat  in  overcoming  resistances. 

The  object  of  the  discussion  of  flow  in  pipes  is  to  enable  the 
discharge  which  will  occur  under  given  conditions  to  be  deter- 
mined, or  to  ascertain  the  proper  size  which  a  pipe  should 
have  in  order  to  deliver  a  given  discharge.  The  subject  can- 
not, however,  be  developed  with  the  defmiteness  which  char- 
acterizes the  flow  from  orifices  and  weirs,  partly  because  the 


1  64  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

condition  of  the  interior  surface  of  the  pipe  greatly  modifies 
the  discharge,  partly  because  of  the  lack  of  experimental  data, 
and  partly  on  account  of  defective  theoretical  knowledge  re- 
garding the  laws  of  flow.  In  orifices  and  weirs  errors  of  two 
or  three  per  cent  may  be  regarded  as  large  with  careful  work  ; 
in  pipes  such  errors  are  common,  and  are  generally  exceeded 
in  most  practical  investigations.  It  fortunately  happens,  how- 
ever, that  in  most  cases  of  the  design  of  systems  of  pipes  errors 
of  five  and  ten  per  cent  are  not  important,  although  they  are 
of  course  to  be  avoided  if  possible,  or,  if  not  avoided,  they 
should  occur  on  the  side  of  safety. 

Prob.  92.  A  pipe  500  feet  long  and  3  inches  in  diameter  dis- 
charges about  48  gallons  per  minute  under  a  head  of  4  feet. 
Compute  the  coefficient  of  discharge. 

ARTICLE  73.  Loss  OF  HEAD  AT  ENTRANCE. 

The  loss  of  head  which  occurs  in  the  upper  end  of  the  pipe, 
due  to  contraction  and  resistance  of  the  inner  edges,  is  called 
the  loss  at  entrance,  and  this  is  the  same  as  in  a  short  cylin- 
drical tube  under  the  same  velocity  of  flow.  Let  c  be  the 
coefficient  of  velocity  or  discharge  for  a  short  tube  and  v  the 
mean  velocity  at  its  outer  end,  then  (Art.  66)  the  loss  of  head 
in  the  tube  is 


Now  this  velocity  v  is  the  same  as  that  in  the  pipe  into  which 
the  tube  may  be  regarded  as  discharging,  and  hence  this  same 
expression  is  the  loss  of  head  which  occurs  at  the  entrance  of 
the  pipe,  or  rather  it  is  the  loss  at  the  upper  end  in  a  length 
equal  to  about  three  diameters. 

The  discussions  of  the  last  chapter  show  that  the  mean 
value  of  c  is  about  0.72  when  the  tube  projects  into  the  reser- 
voir, about  0.82  when  the  inner  end  is  flush  with  side  of  the 


ART.  73.]  LOSS  OF  HEAD  AT  ENTRANCE.  165 

reservoir  and  has  square  corners,  and  that  it  may  be  nearly  i.oo 
when  the  inner  end  is  provided  with  a  bell-shaped  mouth. 
Accordingly  the  loss  of  head  for  a  pipe  projecting  into  the 

reservoir  is 

v*  v* 


and  for  a  pipe  whose  end  is  arranged  like  a  standard  tube, 

/    I  \v*  v* 

f     h'  =  (  —  r-a  —  i  )  —  =  0.49  —  ; 

\0.82a          1  2g  ^  2g 

and  for  a  pipe  with  a  perfect  mouthpiece, 

„=(!,-,  )J=,      • 

The  loss  of  head  at  entrance  is  hence  always  less  than  the 
velocity-head,  and  it  may  be  expressed  by  the  formula 

v* 


in  which  m  is  0.93  for  the  inward  projecting  pipe,  0.49  for  the 
standard  end,  and  o  for  a  perfect  mouthpiece.  When  the  con- 
dition of  the  end  is  not  specified  the  value  used  for  m  in  the 
following  pages  will  be  0.5,  which  supposes  that  the  arrange- 
ment is  like  the  standard  tube  or  nearly  so.  For  short  pipes, 
however,  it  may  be  necessary  to  consider  the  particular  condi- 
tion of  the  end,  and  then 

•  •  •  •  •  •  (49)' 


in  which  c  is  to  be  selected  from  the  evidence  presented  in  the 
last  chapter. 

It  should  be  noted  that  the  loss  of  head  at  entrance  is  very 
small  for  long  pipes.  For  example  it  is  proved  by  actual 
gaugings  that  a  pipe  10  ooo  feet  long  and  i  foot  in  diameter 
discharges  about  4^  cubic  feet  per  second  under  a  head  of  100 
feet.  The  mean  velocity  then  is 

4.25 

v  =  -     —  =  5.41  feet  per  second, 
0.7854      3  ' 


1  66  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

and  the  probable  loss  of  head  at  entrance  hence  is 

Jj  —  o.£  X  0.01555  X  5-4i2  =  0.228  feet, 

or  only  one-fourth  of  one  per  cent  of  the  total  head.  In  this 
case  the  effective  velocity-head  of  the  issuing  stream  is  only 
0.455  feet>  which  shows  that  the  total  loss  of  head  is  99.545 
feet. 

Prob.  93.  Under  a  head  of  20  feet  a  pipe  I  inch  in  diameter 
and  100  feet  long  discharges  15  gallons  per  minute.  Compute 
the  loss  of  head  at  entrance. 

ARTICLE  74.  Loss  OF  HEAD  IN  FRICTION. 

The  loss  of  head  due  to  the  resisting  friction  of  the  interior 
surface  of  a  pipe  is  usually  large,  arid  in  long  pipes  it  becomes 
very  great,  so  that  the  discharge  is  but  a  small  percentage  of 
that  due  to  the  head.  Let  h  be  the  total  head  on  a  pipe  with 

v* 
full  flow,  —  the  velocity-head  of  the  issuing  stream,  h'  the  head 

lost  at  entrance,  and  h"  the  head  lost  in  frictional  resistances. 
Then  if  the  pipe  be  straight  and  of  uniform  size,  so  that  no 
other  losses  occur,  3 

/*  =  —  +  h'  +  h". 

2g^ 

Inserting  for  h'  its  value  from  Art.  73,  this  equation  becomes 


which  is  a  fundamental  formula  for  the  discussion  of  flow  in 
pipes. 

The  head  lost  in  friction  may  be  determined  for  particular 
cases  by  measuring  the  head  h,  the  area  a  of  the  cross-section 
of  the  pipe,  and  the  discharge  per  second  q.  Then  q  divided 
by  a  gives  the  mean  velocity  v,  and  from  the  above  equation 


ART.  74.]  LOSS  OF  HEAD  IN  FRICTION.  1  67 

which  serves  to  compute  //",  the  value  of  c  being  first  selected 
according  to  the  condition  of  the  end.  This  method  is  not 
applicable  to  very  short  pipes  because  of  the  uncertainty 
regarding  the  coefficient  c  (Art.  65). 

Another  method,  and  the  one  most  generally  employed,  is 
by  the  use  of  piezometers  (Art.  70).  A  portion  of  the  pipe 
being  selected  which  is  free  from  sharp  curves,  two  vertical 
tubes  are  inserted  into  which  the  water  rises.  The  differ- 
ence of  level  of  the  water  surfaces  in  the  piezometers  is  then 
the  head  lost  in  the  pipe  between  them,  and  this  loss  is  caused 
by  friction  alone  if  the  pipe  be  straight  and  of  uniform  size.- 

By  these  methods  many  experiments  have  been  made  upon 
pipes  of  different  sizes  and  lengths  under  different  velocities  of 
flow,  and  the  discussion  of  these  has  enabled  the  approximate 
laws  to  be  deduced  which  govern  the  loss  of  head  in  friction, 
and  tables  to  be  prepared  for  practical  use.  These  laws  are  : 

1.  The  loss  in  friction  is  proportional  to  the  length  of  the 

pipe. 

2.  It  increases  nearly  as  the  square  of  the  velocity. 

3.  It  decreases  as  the  diameter  of  the  pipe  increases. 

4.  It  increases  with  the  roughness  of  the  interior  surface. 

5.  It  is  independent  of  the  pressure  of  the  water. 

These  laws  may  be  expressed  by  the  equation 


in  which  /  is  the  length  of  the  pipe,  d  its  diameter,  and  /  is  a 
quantity  which  depends  upon  the  degree  of  roughness  of  the 
surface.  This  equation  is  an  empirical  one  merely  ;  the  theo- 
retic expression  for  h"  is  as  yet  unknown,  and  it  is  probable 
that  when  discovered  it  will  prove  to  be  of  a  complex  nature. 

The  values  of  h"  having  been  deduced  for  a  number  of 


1 68 


FLOW   THROUGH  PIPES. 


[CHAP.  VIL 


cases  in  the  manner  just  explained,  the  corresponding  values 
of /can  be  computed.  In  this  manner  it  is  found  that /varies 
not  only  with  the  roughness  of  the  interior  surface  of  the  pipe, 
but  also  with  its  diameter,  and  with  the  velocity  of  flow.  From 
the  discussions  of  FANNING,  SMITH,  and  others,  the  following 
table  of  mean  values  of  /  has  been  compiled,  which  are  appli- 
cable to  clean  iron  pipes,  either  smooth  or  coated  with  coal- 
tar  varnish,  and  laid  with  close  joints. 

TABLE  XVI.     FRICTION   FACTORS   FOR   PIPES. 


Diameter 

Velocity  in  Feet  per  Second. 

in 
Feet. 

i. 

2. 

3- 

4- 

6. 

10. 

15- 

0.05. 

0.047 

0.041 

0.037 

0.034 

O.O3I 

O.O29 

0.028 

O.  I 

.038 

.032 

.030 

.028 

.026 

.024 

.023 

0.25 

.032 

.028 

.026 

.025 

.024 

.022 

.021 

0.5 

.028 

.026 

.025 

.023 

.022 

.020 

.Oig 

0-75 

.026 

.025 

.024 

.022 

.O2I 

.019 

.018 

I. 

.025 

.024 

.023 

.022 

.020 

.018 

.017 

1.25 

.024 

•023 

.022 

.O2I 

.Oig 

.017 

.016 

i-5 

.023 

.022 

.021 

.020 

.018 

.016 

.015 

1-75 

.022 

.021 

.020 

.018 

.017 

.015 

.014 

2. 

.021 

.020 

.019 

.017 

.016 

.014 

.013 

2-5 

.O2O 

.Oig 

.018 

.016 

.015 

.013 

.012 

3- 

.019 

.018 

.016 

•  015 

.014 

.013 

.012 

3-5 

.018 

.017 

.016 

.014 

.013 

.012 

4- 

.017 

.016 

.015 

.013 

.OI2 

.Oil 

5- 

.Ol6 

.015 

.014 

•  013 

.012 

6. 

.015 

.014 

.013 

.012 

.Oil 

The  quantity /may  be  called  the  friction  factor,  and  the 
table  shows  that  its  value  ranges  from  0.05  to  o.oi  for  new 
clean  pipes.  A  rough  mean  value,  often  used  in  approximate 
computations,  is 

Friction  factor  /  =  0.02. 


ART.  74.]  LOSS   OF  HEAD  IN  FRICTION.  169 

It  is  seen  that  the  tabular  values  of  /  decrease  both  when  the 
diameter  and  when  the  velocity  increases,  and  that  they  vary 
most  rapidly  for  small  pipes  and  low  velocities.  The  probable 
error  of  a  tabular  value  of  f  is  liable  to  be  about  one  unit  in 
the  third  decimal  place,  which  is  equivalent  to  an  uncertainty 
of  ten  per  cent  when  /  =  o.ou,  and  to  five  per  cent  when 
f—  0.021.  The  effect  of  this  is  to  render  computed  values  of 
h"  liable  to  the  same  uncertainties;  but  the  effect  upon  com- 
puted velocities  and  discharges  is  much  less,  as  will  be  seen 
in  Art.  76. 

To  determine,  therefore,  the  probable  loss  of  head  in  fric- 
tion, the  velocity  v  must  be  known,  and  /is  taken  from  the 
table  for  the  given  diameter  of  pipes.  The  formula 

**£& 

J  d'  2g 

then  gives  the  probable  loss  of  head  in  friction.  For  example, 
let  /  =  10  ooo  feet,  d=  i  foot,  v  =  5.41  feet.  Then,  from  the 
table, /is  0.021,  and 

h"  =  0.021  x  10^00  x  0.455  =  96  feet, 

which  is  to  be  regarded  as  an  approximate  value,  liable  to  an 
uncertainty  of  five  per  cent. 

The  theory  of  the  internal  frictional  resistances,  as  far  as 
understood,  indicates  that  the  energy  which  is  thus  transformed 
into  heat  is  expended  in  two  ways :  first,  in  the  direct  friction 
along  the  interior  surface ;  and  second,  in  impact  caused  by  an 
unsteady  motion  of  the  particles  of  water.  Under  very  low 
velocities  the  motion  is  in  lines  parallel  to  the  axis  of  the  pipe, 
so  that  resistance  is  met  only  along  the  surface,  but  under  ordi- 
nary conditions  the  motion  of  many  of  the  particles  is  sinuous, 
whereby  internal  friction  or  impact  is  also  produced.  Experi- 
ments devised  by  REYNOLDS  enable  this  sinuous  motion  to  be 
actually  seen,  so  that  its  existence  is  beyond  question. 


170  FLO  W   THROUGH  PIPES.  [CHAP.  VII. 

Prob.  94.  Determine  the  actual  loss  of  head  in  friction  from 
the  following  experiment  :  /  =  60  feet,  h  —  8.33  feet,  d  = 
0.0878  feet,  q  —  0.03224  cubic  feet  per  second,  and  c  =  0.8. 
Compute  by  help  of  the  table  the  probable  loss  for  the  same 
data. 

ARTICLE  75.  OTHER  LOSSES  OF  HEAD. 

Thus  far  the  pipe  has  been  supposed  to  be  straight  and  o\ 
uniform  size,  so  that  no  losses  of  head  occur  except  at  en- 
trance and  in  friction.  But  if  the  pipe  vary  in  diameter,  or 
have  sharp  curves,  or  contain  valves,  further  losses  occur,  which 
are  now  to  be  considered. 

Sudden  enlargements  and  contractions  of  section  cause 
losses  of  head  which  may  be  ascertained  by  the  rules  of  Arts. 
68  and  69.  These  are  of  infrequent  occurrence  in  pipes,  the 
usual  method  of  passing  from  one  size  to  another  being  by 
means  of  a  "  reducer,"  which  is  a  conical  frustum  several  feet 
long,  whereby  the  velocity  is  slowly  changed  without  expend- 
ing energy  in  impact. 

The  loss  of  head  caused  by  easy  curves  is  very  slight,  and 
need  not  be  taken  into  account.  For  sharp  curves  the  loss  is 
small,  rarely  exceeding  twice  the  velocity-head  for  a  single 
curve,  but  when  many  such  curves  occur  the  item  of  loss  thus 
caused  may  be  important.  According  to  the  investigations  of 
WEISBACH,  the  loss  of  head  due  to  a  curve  of  one-fourth  of  a 
circle  may  be  written 


in  which  n  is  a  number  whose  value  is  given  below  for  different 
values  of  —  ™  where  R  is  the  radius  of  the  curve  of  the  centre 

line  of  the  pipe,  and  d  is  its  diameter  : 

\d 
For  -^  —  o,  o.i,    0.3,    0.4,    0.5,    0.6,    0.7,    0.8,    0.9,    i.o, 

n  =  o,  0.13,  0.16,  0.21,  0.29,  0.44,  0.66,  0.98,  1.41,  1.98 


ART.  75.] 


OTHER  LOSSES   OF  HEAD. 


171 


These  coefficients,  however,  were  derived  for  small  pipes,  and 
it  is  probable  that  for  large  pipes  the  loss  of  head  may  be  less 
than  they  indicate. 

In  Fig.  55  are  shown  three  kinds  of  valves  for  regulating 
the  flow  in  pipes :  at  A  a  valve  consisting  of  a  vertical  sliding- 
gate,  at  B  a  cock-valve  formed  by  two  rotating  segments,  and 


T 


FIG.  55. 


at  C  a  throttle-valve  or  circular  disk  which  moves  like  a  damper 
in  a  stove-pipe.  The  loss  of  head  due  to  these  may  be  very 
large  when  they  are  sufficiently  closed  so  as  to  cause  a  sudden 
change  in  velocity.  It  may  be  expressed  by 


in  which  n  has  the  following  values,  as  determined  by  the  ex- 
periments of  WEISBACH.*  For  the  sluice-valve  let  d'  be  the 
vertical  distance  that  the  gate  is  lowered  below  the  top  of  the 
pipe  ;  then 


For—  =   o 
a 


-1-4-14-4          II 

4  o  Js  o  48 

n  —  o.o    0.07    0.26     0.81     2.1      5.5        17       98 


For   the   cock-valve  let   B  be  the  angle  through  which  it    is 
turned,  as  shown  in  the  figure  ;  then 

For  B  =  o°      10°     20°     30°    40°     50°     55°     60°    65° 
n  =  o      0.29    1.6      5.5       17      53     106     206    486 

*  Mechanics  of  Engineering,  vol.  i.,  COXE'S  translation,  p.  902. 


FLOW   THROUGH  PIPES.  [CHAP.  V1L 

In  like  manner,  for  the  throttle-valve  the  coefficients  are : 
For  6=      5°      10°     20°     30°     40°     50°     60°     65°     ;o° 
72  =  0.24    0.52     1.5     3.9       ii      33      118     256   750 

The  number  n  hence  rapidly  increases  and  becomes  infinity 
when  the  valve  is  fully  closed,  but  as  the  velocity  is  then  zero 
there  is  no  loss  of  head.  The  velocity  v  here,  as  in  other 
cases,  refers  to  that  in  the  main  part  of  the  pipe,  and  not  to 
that  in  the  contracted  section  formed  by  the  valve. 

An  accidental  obstruction  in  a  pipe  may  be  regarded  as 
causing  a  sudden  change  of  section,  and  the  loss  of  head  due 
to  it  is,  by  Art.  68, 


*"-(*-  •)•£-" 


where  a  is  the  area  of  the  section  of  the  pipe,  and  a'  that  of 
the  diminished  section.  This  formula  shows  that  when  a'  is 
one-half  of  #,  the  loss  of  head  is  equal  to  the  velocity-head, 
and  that  n  rapidly  increases  as  a'  diminishes. 

In  the  following  pages  the  symbol  h!"  will  be  used  to 
denote  the  sum  of  all  the  losses  of  head  due  to  curvature* 
valves,  and  contractions  of  section.  Then 


in  which  n  will  denote  the  sum  of  all  the  coefficients  due  to 
these  causes.  In  case  no  mention  is  made  regarding  these 
sources  of  loss  they  are  supposed  not  to  exist,  so  that  both  n 
and  h'"  are  simply  zero. 

Prob.  95.  Compute  for  the  data  of  the  last  problem  the  loss 
of  head  caused  by  a  curve  whose  radius  is  2  feet. 

Ans.  0.013  feet. 


ART.  76.]  FORMULA   FOR    VELOCITY.  1/3 

ARTICLE  76.  FORMULA  FOR  VELOCITY. 

The  mean  velocity  in  a  pipe  can  now  be  deduced  for  the 
condition  of  full  flow.     The  total  head  being  h,  and  the  effec- 

v* 
tive  velocity-head  of  the  issuing  stream  being  —  ,  the  lost  head 

tf 
is  h  --  ,  and  this  must  be  equal  to  the  sum  of  its  parts,  or 


(52) 


Substituting  in  this  the  values  of  //,  /*",  and  h'"  from  the  pre- 
ceding articles,  it  becomes 

V*  V*  I    V*  V* 

h  --  =m  --  (-/-T  --  \-n—  ;     .     .     (52)' 
2g  2g       '  d  2g          2g 

and  by  solving  for  v  there  is  found 


which  is  a  general  formula  for  the  velocity  of  flow. 

In  this  formula  n  will  be  taken  as  o,  unless  otherwise  stated  ; 
that  is,  no  losses  of  head  occur  except  at  entrance  and  in  fric- 
tion. The  formula  for  pipes  which  are  essentially  straight  and 
of  uniform  size  throughout  then  is 


Here  m  is  taken  as  0.5,  which  is  to  be  regarded  as  its  mean 
value  in  accordance  with  the  discussion  in  Art.  73. 

In  this  formula  the  friction  factor/"  is  a  function  of  v  to  be 
taken  from  the  table  in  Art.  74,  and  hence  v  cannot  be  directly 


7, 


174  FLOW   THROUGH  PIPES.  [CHAP.  VIL 

computed,  but  must  be  obtained  by  successive  approximations. 
For  example,  let  it  be  required  to  compute  the  velocity  of  dis~ 
charge  from  a  pipe  3000  feet  long  and  6  inches  in  diameter 
under  a  head  of  9  feet.  Here  /  =  3000,  d  —  0.5,  and  h  =  9  ; 
taking  for /the  rough  mean  value  0.02,  the  formula  gives 


2  X  32.16  x  9 


[-5  +  0.02  X  3000  X  2 

The  approximate  velocity  is  hence  2.2  feet  per  second,  and 
entering  the  table  with  this,  the  value  of /is  found  to  be  0.026. 
Then  the  formula  gives 

2  X  32.16  X  9          __ 
[.5  +  0.026  X  3000  X  2  ~ 

This  is  to  be  regarded  as  the  probable  value  of  the  velocity, 
since  the  table  gives  /  =  0.026  for  v  =  1.92.  In  this  manner 
by  one  or  two  trials  the  value  of  v  can  be  computed  so  as  to 
agree  with  the  corresponding  value  of/ 

The  error  in  the  computed  velocity  due  to  an  error  of  one 
unit  in  the  last  decimal  of  the  factor  /  is  always  relatively  less 
than  the  error  in  /  itself.  For  instance,  if  v  be  computed  for 
the  above  example  with/=  0.025,  its  value  is  found  to  be  1.96 
feet  per  second,  or  two  per  cent  greater  than  1.92.  In  general, 
the  percentage  of  error  in  v  is  less  than  one-half  of  that  in  / 
It  hence  appears  that  computed  velocities  are  liable  to  probable 
errors  ranging  from  one  to  five  per  cent,  owing  to  imperfections 
in  the  tabular  values  of  /  for  new  clean  pipes.  This  uncer- 
tainty is  as  a  rule  still  further  increased  by  various  causes,  so 
that  five  per  cent  is  to  be  regarded  as  a  common  probable  error 
in  computations  of  velocity  and  discharge  from  pipes. 

Velocities  greater  than  15  feet  per  second  are  very  unusual 
in  pipes,  and  but  little  is  known  as  to  the  values  of/  for  such 
cases.  For  velocities  less  than  one  foot  per  second,  the  values 


ART.  77-]  COMPUTATION  OF  DISCHARGE.  1/5 

of /are  also  not  understood,  so  that  little  reliance  can  be  placed 
upon  computations.  The  usual  velocity  in  water  mains  is  less 
than  five  feet  per  second,  it  being  found  inadvisable  to  allow 
swifter  flow  on  account  of  the  great  loss  of  head  in  friction. 

To  illustrate  the  use  of  the  general  formula,  let  the  pipe  in 
the  above  example  be  supposed  to  have  a  curve  of  6  inches 
radius,  and  to  contain  a  gate  valve  which  is  half  closed.  Then 
from  Art.  75,  n  —  0.29  for  the  curve  and  n  =  2.1  for  the  valve, 
or  in  the  formula  n  is  to  be  put  as  2.39.  The  velocity  is  now 
found  to  be 


2  X  32-16  X  9  t  j 

=  1.90  feet  per  second  ; 


5.89  +  0.026  x  6ooo 

which  is  but  a  trifle  less  than  that  found  before.  The  closing 
of  the  sluice  gate  to  one-half  its  depth  hence  but  slightly  in- 
fluences the  velocity,  while  the  effect  of  the  curve  is  scarcely 
perceptible.  With  a  shorter  pipe,  however,  the  influence  of 
these  would  be  more  marked. 

Prob.  96.  Compute  the  velocity  for  the  data  of  the  last 
example  if  the  pipe  be  1000  feet  long. 

Prob.  97.  Compute  the  velocity  for  a  pipe  15  ooo  feet  long 
and  1 8  inches  in  diameter  under  a  head  of  230  feet. 

Ans.  9.57  feet  per  second. 

ARTICLE  77.  COMPUTATION  OF  DISCHARGE. 

The  discharge  per  second  from  a  pipe  of  given  diameter  is 
found  by  multiplying  the  velocity  of  discharge  by  the  area  of 
the  cross-section  of  the  pipe,  or 

in  which  v  is  to  be  found  by  the  method  of  the  last  article. 

For  example,  let  it  be  required  to  find  the  discharge  in 
gallons  per  minute  from  a  clean  pipe  3  inches  in  diameter  and 


FLOW   THROUGH  PIPES.  [CHAP.  VII. 

500  feet  long  under  a  head  of  4  feet.  Here  d  =  0.25,  /  =  500, 
and  h  =  4.  Then  for/=  0.02,  the  velocity  is  found  to  be  2.5 
feet  per  second  ;  again  taking  from  the  table  f  =  0.027,  the 
velocity  is  2.15  feet  per  second.  The  discharge  in  cubic  feet 
per  second  is 

q  =  0.7854  X  0.25'  X  2.15  =  0.106; 
and  in  gallons  per  minute, 

q  —  o.i 06  X  7.48  X  60  =  47.6. 

This  is  the  probable  result,  which  is  liable  to  the  same  uncer- 
tainty as  the  velocity — say  about  three  per  cent ;  so  that  strictly 
the  discharge  should  be  written  47.6  ±  1.4  gallons. 

By  inserting  the  value  of  v  in  the  above  expression  for  q  it 
becomes 

q—- 


and  from  this  the  value  of   the  head  required  to  produce  a 
given  discharge  is 


These  formulas  are  not  more  convenient  for  practical  computa- 
tions than  the  separate  expressions  for  v,  q,  and  h  previously 
established,  since  in  any  event  v  must  be  computed  in  order  to 
select  f  from  the  table.  They  serve,  however,  to  exhibit  the 
general  laws  which  govern  the  discharge. 

Prob.  98.  Compute  the  probable  discharge  from  a  pipe   I 
inch  in  diameter  and  1000  feet  longtunder  a  head  of  40  feet. 

Prob.  99.  What  head  is  required  to  discharge  3  gallons  per 
minute  through  a  pipe  I  inch  in  diameter  and  1000  feet  long? 

Ans.  11.3  feet. 


A  RT.  78  .  ]  COMP  UTA  TION  OF  DIA  ME  TER.  1  77 

ARTICLE  78.  COMPUTATION  OF  DIAMETER. 

It  is  an  important  practical  problem  to  determine  the 
diameter  of  a  pipe  to  discharge  a  given  quantity  of  water 
under  a  given  head  and  length.  The  last  equation  above 
serves  to  solve  this  case,  as  all  the  quantities  in  it  except  d 
are  known.  This  may  be  written  in  the  form 


or  placing  for  m  and  2g  their  mean  values  and  neglecting  n,  it 
becomes 


(55) 


which  is  the  formula  for  computing  d  when  ^,  /,  and  d  are  in 
feet  and  q  is  in  cubic  feet  per  second.  The  value  of  the  fric- 
tion factor  /may  be  taken  as  0.02  in  the  first  instance,  and  the 
d  in  the  right-hand  member  being  neglected,  an  approximate 
value  of  the  diameter  is  computed.  The  velocity  is  next 
found  by  the  formula 

q 


v  = 


a  ' 


and  from  Table  XVI.  the  value  of  f  thereto  corresponding  is 
selected.  The  computation  for  d  is  then  repeated,  placing  in 
the  right-hand  member  the  approximate  value  of  d.  Thus 
by  one  or  two  trials  the  diameter  is  computed  which  will 
satisfy  the  given  conditions. 

For  example,  let  it  be  required  to  determine  the  diameter 
of  a  pipe  which,  under  the  condition  of  full  flow,  will  deliver 
500  gallons  per  second,  its  length  being  4500  feet  and  the 
head  24  feet.  Here  the  value  of  q  is 

q  =  — —  =  66.84  cubic  feet. 
7.481 


I78  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

The  approximate  value  of  d  then  is 

/o.02  x  4500  X  66.842U 
d  =  o.479(-          ±^—          -f  -  3-35  feet. 

From  this  the  velocity  of  flow  is 

66.84 

v  =  — „ —      5  =  7.6  feet  per  second, 

0.7854  X  3-35 

and  from  the  table  the  value  of /for  this  diameter  and  velocity 
is  found  to  be  0.013.  Then 

d  =  0.479  [(1-5  X  3-35  +  0.013  X  4500)  ~£^J\ 

from  which  d  =  3.125  feet.  With  this  value  of  d  the  velocity 
is  now  found  to  be  8.71  feet,  so  that  no  change  results  in  the 
value  of  f.  The  required  diameter  of  the  pipe  is  therefore 
3.1  feet,  or  about  37  inches;  but  as  the  regular  market  sizes  of 
pipes  furnish  only  36  inches  and  40  inches,  one  of  these  must 
be  used,  and  it  will  be  on  the  side  of  safety  to  select  the 
larger. 

It  will  be  well  in  determining  the  size  of  a  pipe  to  also  con- 
sider that  the  interior  surface  may  become  rough  by  erosion 
and  incrustation,  thus  increasing  the  value  of  the  friction  fac- 
tor and  diminishing  the  discharge.  The  increase  in  f  from 
these  causes  is  not  likely  to  be  so  great  in  a  large  pipe  as  in  a 
small  one,  but  it  is  thought  that  for  the  above  example  they 
might  be  sufficient  to  make  f  as  large  as  0.03.  Applying  this 
value  to  the  computation  of  the  diameter  from  the  given  data 
there  is  found  d  =  3.6  feet  =  about  43  inches. 

The  sizes  of  pipes  generally  found  in  the  market  are  -J-,  f r 
i,  lib  !i»  2>  3,  4>  6,  8,  10,  12,  1 6,  1 8,  20,  24,  27,  30,  36,  40,  44, 
and  48  inches,  while  intermediate  or  larger  sizes  must  be  made 
to  order.  The  computation  of  the  diameter  is  merely  a  guide 
to  enable  one  of  these  sizes  to  be  selected,  and  therefore  it 


ART.  79.]  SHORT  PIPES.  1/9 

is  entirely  unnecessary  that  the  numerical  work  should  be  car- 
ried to  a  high  degree  of  precision.  In  fact,  three-figure  loga- 
rithms are  usually  sufficient  to  determine  reliable  values  of  d. 

Prob.  100.  Compute  the  diameter  of  a  pipe  to  deliver  50 
gallons  per  minute  under  a  head  of  4  feet  when  its  length  is 
500  feet.  Also  when  its  length  is  5000  feet. 

ARTICLE  79.    SHORT  PIPES. 

A  pipe  is  said  to  be  short  when  its  length  is  less  than  about 
500  times  its  diameter,  and  very  short  when  the  length  is  less 
than  about  50  diameters.  In  both  cases  the  coefficient  c  should 
be  estimated  according  to  the  condition  of  the  upper  end  as 
precisely  as  possible,  and  the  length  /  should  not  include  the 
first  three  diameters  of  the  pipe,  as  that  portion  properly  be- 
longs to  the  tube  which  is  regarded  as  discharging  into  the 
pipe.  In  attempting  to  compute  the  discharge  for  such  pipes, 
it  is  often  found  that  the  velocity  is  greater  than  given  in 
Table  XVI.,  and  hence  that  the  friction  factor  f  cannot  be  de- 
termined. For  this  reason  no  accurate  estimate  can  be  made 
of  the  discharge  from  short  pipes  under  high  heads,  and  fortu- 
nately it  is  not  often  necessary  to  use  them  in  engineering 
constructions. 

For  example,  let  it  be  required  to  compute  the  velocity  of 
flow  from  a  pipe  i  foot  in  diameter  and  100  feet  long  under  a 
head  of  100  feet,  the  upper  end  being  so  arranged  that  c  =  0.80, 
and  hence  m  =0.56  (Art.  73).  Neglecting  n,  since  the  pipe 
has  no  curves  or  valves,  the  formula  for  the  velocity  becomes 


v  = 


and  using  for  f  the  rough  mean  value  O.O2, 


/ 

v  =  V  1. 


64.32  x  100 

=  42'9 


0.02x97 


180  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

Now  there  is  absolutely  no  experimental  knowledge  regarding 
the  value  of  the  friction  facfor  f  for  such  high  velocities,  but 
judging  from  the  table  it  is  probable  that  /  may  be  about 
0.015.  Using  this  instead  of  0.02  gives  for  v  the  value  46  feet 
per  second.  The  uncertainty  of  this  result  should  be  regarded 
as  at  least  ten  per  cent. 

For  very  short  pipes  there  are  on  record  a  number  of  ex- 
periments by  EYTELWEIN  and  others,  from  which  the  coeffi- 
cients of  discharge  have  been  deduced.  The  upper  end  of  the 
pipe  being  in  all  cases  arranged  like  the  standard  tube,  these 
experiments  give  the  following  as  mean  values  of  the  velocity : 


For 

7    

34 

v 

=  0.82 

V2gk 

For 

/    = 

\2d, 

V 

=  0.77 

\f2gk 

For 

/    = 

24d, 

V 

=  0.73 

V2jk 

For 

/    = 

36^ 

V 

=  0.68 

^2gk 

For 

1    = 

484 

V 

=  0.63 

\^k 

For 

1    

6od, 

V 

=  0.60 

V2gh 

These  coefficients  were  deduced  for  small  pipes  under  low 
heads,  and  are  to  be  regarded  as  liable  to  a  variation  of  several 
per  cent ;  for  large  pipes  and  high  heads  they  are  all  probably 
too  large. 

The  general  equation  for  the  velocity  of  discharge  deduced 
in  Art.  76  may  be  applied  to  very  short  pipes  by  writing  /  —  $d 
in  place  of  /,  and  placing  for  m  its  value  in  terms  of  c.  It  then 
becomes 


_  /  2gh 


If  in  this  /  equals  3^,  the  velocity  is 

V  =  C 


ART.  80.]  LONG  PIPES.  l8l 

which  is  the  same  as  for  a  short  tube.  If  /  =  I2</,  f  =  0.02, 
and  c  =  0.82,  it  gives  v  =  0.774  V2gk,  which  agrees  well  with 
the  mean  value  above  stated. 

Prob.  101.  Compute  the  discharge  per  second  for  a  pipe 
I  inch  in  diameter  and  40  inches  long  under  a  head  of  4  feet. 

ARTICLE  80.  LONG  PIPES. 

For  long  pipes  the  loss  of  head  at  entrance  becomes  very 
small  compared  with  that  lost  in  friction,  and  the  velocity-head 
is  also  small.  The  formula  for  velocity  deduced  in  Art.  76  is 

2gh 


•••>+4 

in  which  the  first  term  in  the  denominator  represents  the  effect 
of  the  velocity-head  and  the  entrance-head,  the  mean  value  of 
the  latter  being  0.5.  Now  it  may  safely  be  assumed  that  1.5 
may  be  neglected  in  comparison  with  the  other  term,  when  the 
error  thus  produced  in  v  is  less  than  one  per  cent.  Taking  for 
f  its  mean  value,  this  will  be  the  case  when 


V 


I 

=  i.oi,     from  which    -^  =  3750. 


°-02d  i 


Therefore  when  /  is  greater  than  about  4000^  the  pipe  will 
be  called  long. 

For  long  pipes  the  velocity  under  full  flow  hence  is  given 
by  the  formula 

A/2gdh  i/  dh 

v  =  V^r  =  %jo2Vji,       ....    (57) 

and  the  discharge  per  second  is, 

v  =  6.30  A /4£ (57)' 


J82  FLO  W   THROUGH  PIPES.  [CHAP.  VII. 

For  computing  the  diameter  required  to  deliver  a  given  dis- 
charge the  formula  is 

d=  0.479  (^f) (57)" 

These  equations  show  that  for  very  long  pipes  the  discharge 
varies  directly  as  the  2-J-  power  of  the  diameter,  and  inversely 
as  the  square  root  of  the  length. 

In  the  above  formulas  d,  h,  and  /  are  to  be  taken  in  feet,  q  in 
cubic  feet  per  second,  and  f  is  to  be  found  from  the  table  in 
Art.  74,  an  approximate  value  of  v  being  first  obtained  by 
taking /as  0.02.  It  should  not  be  forgotten  that  these  expres- 
sions are  of  an  empirical  nature,  and  do  not  necessarily  rep- 
resent the  true  laws  of  flow  ;  but  at  present  they  seem  to  be  the 
representation  of  these  laws  which  for  long  pipes  best  agrees 
with  experiments.  The  value  of  h  in  these  formulas  is  also 
really  the  friction-head  h",  since  in  their  deduction  the  other 

heads,  h ',  hf",  and  — ,  have  been  neglected  ;  these,  however,  al- 

o 

though  often  very  small,  can  never  be  really  zero. 

Prob.  102.  Compute  the  probable  discharge  from  a  pipe 
26500  feet  long  and  18  inches  in  diameter  under  a  head  of 
324.7  feet.  Ans.  14.7  cubic  feet  per  second. 

Prob.  103.  Compute  the  diameter  required  to  deliver  15  ooo 
cubic  feet  per  hour  through  a  pipe  26  500  feet  long  under  a 
head  of  324.7  feet.  If  this  quantity  is  carried  in  two  pipes  of 
equal  diameter,  what  should  be  their  size? 

ARTICLE  81.  RELATIVE  DISCHARGING  CAPACITIES. 

For  orifices  and  short  tubes  the  discharge  under  a  given 
head  varies  as  the  square  of  the  diameter.  In  pipes  of  equal 
length  under  given  heads  the  discharges  vary  more  rapidly 


ART.  Si.]          RELATIVE  DISCHARGING  CAPACITIES.  1&3 

than  the  squares  of  the  diameters,  owing  to  the  influence  of  fric- 
tion.    For  a  long  pipe  the  formula  for  discharge  is 


q  =  i 

which  shows  that  if  f  be  constant  the  discharge  varies  as  the 
2^  power  of  the  diameter.  This  is  a  useful  approximate  rule 
for  comparing  the  relative  discharging  capacities  of  pipes. 

Thus  if  there  be  two  pipes  with  diameters  dl  and  d^  the  rule 
gives 

q,:q*  =  dtidj,    ....     .     .-.(58) 

and  from  this 


(58)' 


For  example,  if  there  be  two  pipes  of  the  same  length  under 
the  same  head,  the  first  one  foot  and  the  second  two  feet  in 
diameter, 

2\l 


or  the  second  pipe  discharges  nearly  six  times  as  much  as  the 
first.  In  other  words,  six  pipes  of  I  foot  diameter  are  about 
equivalent  to  one  pipe  of  2  feet  diameter. 

As  the  friction  factor  /is  not  constant,  the  above  rule  is  not 
exact  ;  for,  as  the  formula  shows, 


from  which 

?'  =  9>  (-JJ  (A 

Now  as  the  values  of  f  vary  not  only  with  the  diameter  but 
with  the  velocity,  a  solution,  cannot  be  made  except  in  partic- 
ular cases.  For  the  above  example  let  the  velocity  be  about 


1 84  FLOW   THROUGH  PIPES.  [CHAP.  VI L 

3    feet    per    second ;    then    from   the   table  /,  —  0.023    and 

f^  —  0.019,  and 

&  =  &  (2)*  (i. 2)*  =  6.2?,, 

or  the  two-foot  pipe  discharges  more  than  six  times  as  much 
as  the  one-foot  pipe. 

Prob.  104.  How  many  pipes,  6  inches  in  diameter,  are  equiv- 
alent in  discharging  capacity  to  one  pipe  24  inches  in  diameter? 


ARTICLE  82.  A  COMPOUND  PIPE. 

A  compound  pipe  is  one  laid  with  different  sizes  in  differ- 
ent portions  of  its  length.  In  such  the  change  from  one  size 
to  another  is  to  be  made  gradually  by  a  reducer,  so  that  losses 
of  head  due  to  sudden  enlargement  or  contraction  are  avoided 
(Art.  68).  Let  d^  ,  dz  ,  dz  ,  etc.,  be  the  diameters  ;  ^  ,  ^  ,  /3  ,  etc., 
the  corresponding  lengths,  the  total  length  being  /x  +  4  +  etc- 
Let  vlt  v9,  etc.,  be  the  velocities  in  the  different  sections. 
Neglecting  the  loss  of  head  at  entrance,  the  total  head  h  may 
be  placed  equal  to  the  loss  of  head  in  friction,  or 


+etc-  •  •  •  (6o) 

Now  if  the  discharge  per  second  be  g, 

7J    —       *  n    —  —L±—        ptr 

"«/,"  •"*</," 

Substituting  these  and  solving  for  q,  gives 

,     .     .     .     (60)' 


I 
V 

v 


-rr  H-  A  -4r  4-  etc. 


in  which  flf/9,  etc.,  are  the  friction  factors  corresponding  to 
the  given  diameters  and  velocities  in  Table  XVI. 


ART.  82.]  A    COMPOUND  PIPE.  185 

For   example,  take  the  case  of   two   sizes   for  which  the 
dimensions  are 

dl  —  2  feet,  /!  =  2800  feet, 

d*  =  1.5  feet,  /„  =  2145  feet>  ^  =  I27-5  feet. 

Using  for/!  and  /,  the  mean  value  0.02,  and  making  the  sub- 
stitutions %in  the  formula,  there  is  found 

q  =  27.3  cubic  feet  per  second, 
from  which        vl  =  8.7  and  v^  =  15.4  feet  per  second. 

Now  from  the  table  in  Art.  74  it  is  seen  that  ft  =  0.015  and 
/a  =  0.015  ;  and  repeating  the  computation, 

q  —  30.1  cubic  feet  per  second, 
which  gives       vl  =  9.60  and  v^  =  17.0  feet  per  second. 

These  results  are  probably  as  definite  as  the  table  of  friction 
factors  will  allow,  but  are  to  be  regarded  as  liable  to  an  uncer- 
tainty of  several  per  cent. 

To  determine  the  diameter  of  a  pipe  which  will  give  the 
same  discharge  as  the  compound  one,  it  is  only  necessary  to 

replace  the  denominator  in  the  above  value  of  q  by  'f-js>  where 

/  =  /!  +  /.,+  etc.,  and  d  is  the  diameter  required.  Taking  the 
values  of  /as  equal,  this  gives 


Applying  this  to  the  above  example,  it  becomes 

4945   .  .  2800         2145 
d*~    -      25  i.55' 

from  which  d  =  1.68  feet,  or  about  20  inches. 


186  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

Compound  pipes  are*  sometimes  used  in  order  to  prevent 
the  hydraulic  gradient  (Article  84)  from  falling  below  the  pipe. 
At  Rochester,  N.  Y.,  there  is  a  pipe  101  261  feet  long,  of  which 
50  776  feet  is  36  inches  in  diameter,  and  50  485  feet  is  24  inches 
in  diameter.  Under  a  head  of  385.6  feet  this  pipe  discharged 
in  1876  about  14  cubic  feet  per  second  and  in  1890  about  loj 
cubic  feet  per  second. 

Prob.  105.  Compute  the  discharge  of  the  Rochester  pipe, 
using  the  table  on  page  168. 

ARTICLE  83.  PIEZOMETER  MEASUREMENTS. 

Let  a  piezometer  tube    be   inserted   into    a    pipe  at    any 

point  Dl ,  whose  distance 
from  the  reservoir  is  /t 
measured  along  the  pipe 
line.  Let  A1D1  be  the 
vertical  depth  of  this 
point  below  the  water 
level  of  the  reservoir; 
then  if  the  flow  be  stopped 
at  the  end  C,  the  water  rises  in  the  tube  to  the  point  A1 .  But 
when  the  flow  occurs,  the  water  level  in  the  piezometer  stands 
at  some  point  Cl ,  and  the  piezometric  height  or  pressure-head 
is  h^ ,  or  C,D,  in  the  figure.  The  distance  AlCl  then  represents 
the  velocity-head  plus  all  the  losses  of  head  between  Dl  and 
the  reservoir.  If  no  losses  of  head  occur  except  at  entrance 
and  in  friction,  the  value  of  AlCl  then  is 

TT  V*  V*  .r   £      V*  r*    N 

HI  = \-m — -f/_i  —  ,.     .     .     .     (61) 

2g  2g  d    2g 

from  which  the  piezometric  height  can  be  found  when  v  has 
been  determined  by  measurement  or  by  computation. 

For  example,  let  the  total  length  /  =  3000  feet,  d—  6  inches, 
h  =  9  feet,  and  m  ==  0.5.  Then,  as  in  Art.  76,  there  is  found 


ART.  83.]  PIEZOMETER  MEASUREMENTS.  l8/ 

f  —  0.026  and  v  =  1.917  feet  per  second.  The  position  of 
the  top  of  the  piezometric  column  is  then  given  by 

H,  =  (1.5  +0.052/,)  X  0.05714, 
and  the  height  of  that  column  is 
^  =  A,D,-  H,. 

Thus  if  /!  =  1000  feet,  H,  =  3.06  feet ;  and  if  /,  =  2000  feet, 
H^  =  6.03  feet.  If  the  pipe  is  so  laid  that  AlDl  is  9  feet,  the 
corresponding  pressure-heights  are  then  5.94  and  2.97  feet. 

For  a  second  piezometer  inserted  at  Z>,  at  the  distance  /, 
from  the  entrance  the  value  of  H^  is 

ff.=  *-  +  „*+  Av--.  (61)' 

2£-  2g          J   d  2g 

From  this,  subtracting  the  preceding  equation,  there  is  found 

fft-ffl=f<^± (62) 

The  second  member  of  this  formula  is  the  head  lost  in  friction 
in  the  length  /2  —  />  (Art.  74),  and  the  first  member  is  the 
difference  of  the  piezometer  elevations.  Thus  is  again  proved 
the  principle  of  Art.  70,  that  the  difference  of  two  piezometer 
elevations  shows  the  head  lost  in  the  pipe  between  them ;  in 
Art.  70  the  elevations  H,  and  H^  were  measured  upward  from 
the  datum  plane,  while  here  they  are  measured  downward. 

By  the  help  of  this  principle  the  velocity  of  flow  in  a  pipe 
may  be  approximately  determined.  A  line  of  levels  is  run 
between  the  points  Dl  and  D^ ,  which  are  selected  so  that  no 
sharp  curves  occur  between  them,  and  thus  the  difference 
//a  —  Hl  is  found ;  /a  —  /,,  or  the  length  between  D,  and  Z>2, 


1  88  FLO  W   THROUGH  PIPES.  [CHAP.  VII. 

is  ascertained   by  careful  chaining.      Then,    from    the   above 
formula, 


from  which  v  can  be  computed  by  the  help  of  the  friction  fac- 
tors in  the  table  of  Art.  74.  For  example,  STEARNS,  in  1880, 
made  experiments  on  a  conduit  pipe  4  feet  in  diameter  under 
different  velocities  of  flow.*  In  experiment  No.  2  the  length 
/„  —  A  was  1747.2  feet,  and  the  difference  of  the  piezometer 
levels  was  1.243  feet-  Assuming  for  /the  mean  value  0.02,  and 
using  32.16  for^-,  the  velocity  was 

.  764.32  X  1.243  X  4  , 

v  =  \     —  —  -  —  -  -  —  3.0  feet  per  second. 
V         0.02  X  1747 

This  velocity  in  the  table  of  friction  factors  gives/"  =  0.015  for 
a  4-foot  pipe.  Hence,  repeating  the  computation,  there  is 
found  v  =  3.50  feet  per  second  ;  it  is  accordingly  uncertain 
whether  the  value  of  /is  0.015  or  0.014.  If  the  latter  value  be 
used  there  is  found 

v  =  3.62  feet  per  second. 

The  actual  velocity,  as  determined  by  measurement  of  the 
water  over  a  weir,  was  3.738  feet  per  second,  which  shows  that 
the  computation  is  in  error  about  4  per  cent. 

The  gauging  of  the  flow  of  a  pipe  by  piezometers  is  liable 
to  give  defective  results,  partly  because  the  piezometer  may 
not  indicate  the  mean  pressure  in  the  pipe  owing  to  an  imper- 
fect manner  of  connection,  and  partly  because  the  formula  for 
computing  the  velocity  is  merely  an  empirical  one.  The  dif- 
ference ffa  —  Hl  in  order  to  be  reliable  should  be  taken  at 

*  Transactions  of  American  Society  of  Civil  Engineers,  1885,  vol.  xiv.  p.  i. 


ART.  84.]  THE  HYDRAULIC  GRADIENT.  189 

points  as  far  apart  as  possible,  and  care  be  taken  that  no  losses 
of  head  occur  between  them  except  that  due  to  friction.  Easy 
curves  give  no  perceptible  loss  of  head  and  need  not  be  con- 
sidered, but  obstructions  in  the  pipe  or  changes  in  section  may 
render  the  measurement  valueless.  When  pressure  gauges 
are  used,  as  must  be  often  the  case  under  high  heads,  care 
should  be  taken  to  test  them  before  making  the  experiment. 
The  pressure  gauges,  as  generally  graduated,  give  the  pres- 
sures in  pounds  per  square  inch.  If  then  the  readings  pl  and 
/3  are  taken  at  Dl  and  Z>2,  the  pressure-heads  in  feet  are 

Al  =  2.304^     and     At  =  2.304/3. 

The  vertical  distances  AJ)^  and  A  JD^  having  been  previously 
determined  by  levels,  the  heads  Hl  and  H^  are 

H,  =  A,D,  -  h,     and     H,  =  AZDZ  -  h%, 

from  which  Ht  — '  Hv  is  known.  Or  if  the  vertical  fall  z  be- 
tween Dl  and  Z>2  is  determined, 

fft-jrt  =  A, -*,  +  *, 

which  is  the  loss  of  head  between  Dl  and  Z>2 

Prob.  1 06.  At  a  point  500  feet  from  the  reservoir,  and  28 
feet  below  its  surface,  a  pressure  gauge  reads  10.5  pounds  per 
square  inch  ;  at  a  point  8500  feet  from  the  reservoir  and  280.5 
feet  below  its  surface,  it  reads  61  pounds  per  square  inch. 
Show  that  the  discharge  per  second  is  about  6  cubic  feet  if  the 
pipe  be  12  inches  in  diameter. 

ARTICLE'  84.  THE  HYDRAULIC  GRADIENT. 

The  hydraulic  gradient  is  a  line  which  connects  the  water 
levels  in  piezometers  placed  at  intervals  along  the  pipe ;  or 
rather,  it  is  the  line  to  which  the  water  levels  would  rise  if 
piezometer  tubes  were  inserted.  In  Fig.  56  the  line  BC  is  the 


FLOW   THROUGH  PIPES.  [CHAP.  VI  L 

hydraulic  gradient,  and  it  is  now  to  be  shown  that  for  a  pipe 
of  uniform  size  this  is  approximately  a  straight  line.  For  a 
pipe  discharging  freely  into  the  air,  as  in  Fig.  56,  this  line  joins 
the  outlet  end  with  a  point  B  near  the  top  of  the  reservoir. 
For  a  pipe  with  submerged  discharge,  as  in  Fig.  57,  it  joins  the 
lower  water  level  with  the  point  B. 

Let  Dl  be  any  point  on  the  pipe  distant  /,  from  the  reser- 

voir, measured  along  the 
pipe  line.  The  piezometer 
there  placed  rises  to  C19 
which  is  a  point  in  the 
hydraulic  gradient.  The 
equation  of  this  line  with 
FIG.  57  reference  to  the  origin  A  is 

given  by  the  formula  of  the  preceding  article, 


in  which  Hl  is  the  ordinate  A^^  and  /T  is  the  abscissa  AA1}  pro- 
vided that  the  length  of  the  pipe  is  sensibly  equivalent  to  its 

v* 
horizontal  projection.     In  this  equation  the  term  (i  -]-  m)  —  is 

constant  for  a  given  velocity,  and  is  represented  in  the  figure 
by  AB  or  A^Bl  ;  the  second  term  varies  with  /x  ,  and  is  repre- 
sented by  BlCl  .  The  gradient  is  therefore  a  straight  line,  sub- 
ject to  the  provision  that  the  pipe  is  laid  approximately  hori- 
zontal ;  which  is  usually  the  case  in  practice,  since  quite  mate- 
rial vertical  variations  may  exist  in  long  pipes  without  sensibly 
affecting  the  horizontal  distances. 

When  the  variable  point  Dl  is  taken  at  the  outlet  end  of 
the  pipe,  //,  becomes  the  head  //,  and  /,  becomes  the  total 
length  /,  agreeing  with  the  formula  of  Art.  76,  if  the  losses  of 
head  due  to  curvature  and  valves  be  omitted.  When  Dl  is 


ART.  84.]  THE  HYDRAULIC  GRADIENT.  IQI 

taken  very  near  the  inlet  end,  /  becomes  zero  and  the  ordinate 
Hl  becomes  AB,  which  represents  the  velocity-head  plus  the 
loss  of  head  at  entrance. 

When  easy  horizontal  curves  exist,  the  above  conclusions 
are  unaffected,  except  that  the  gradient  BC  is  always  vertically 
above  the  pipe,  and  therefore  can  be  called  straight  only  by 
courtesy,  although  as  before  the  ordinate  BlCl  is  proportional 
to  /, .  When  sharp  curves  exist,  the  hydraulic  gradient  is  de- 
pressed at  each  curve  by  an  amount  equal  to  the  loss  of  head 
which  there  occurs. 

If  the  pipe  is  so  laid  that  a  portion  of  it  rises  above  the  hy- 
draulic gradient  as  at  D1  in  Fig.  58,  an  entire  change  of  condi- 
tion generally  results.  If 
the  pipe  be  closed  at  C,  all 
the  piezometers  stand  in 
the  line  AA,  at  the  same 
level  as  the  surface  of  the 
reservoir.  When  the  valve 
at  C  is  opened,  the  flow  at  FIG.  5s. 

first  occurs  under  normal  conditions,  h  being  the  head  and  BC 
the  hydraulic  gradient.  The  pressure-head  at  JDl  is  then  neg- 
ative, and  represented  by  D:Cr  This  results  in  a  partial  vacu- 
um in  that  portion  of  the  pipe  whereby  the  continuity  of  the 
flow  is  broken,  and  as  a  consequence  the  pipe  from  Dl  to  C  is 
only  partly  filled  with  water.  The  hydraulic  gradient  is  then 
shifted  to  BDl ,  the  discharge  occurs  at  Dl  under  the  head 
AJ}^  while  the  remainder  of  the  pipe  acts  merely  as  a  channel 
to  deliver  the  flow.  It  usually  happens  that  this  change  re- 
sults in  a  great  diminution  of  the  discharge,  so  that  it  has 
often  been  necessary  to  dig  up  and  relay  portions  of  a  pipe  line 
which  have  been  inadvertently  run  above  the  hydraulic  gra- 
dient. This  trouble  can  always  be  avoided  by  preparing  a 


FLOW   THROUGH  PIPES.  [CiiAP.  VJI. 

profile  of  the  proposed  route,  and  drawing  the  hydraulic  gra- 
dient upon  it. 

When  a  large  part  of  a  pipe  lies  above  the  hydraulic  gradi- 
ent it  is  called  a  siphon.  Conditions  sometimes  exist  which 
require  the  construction  of  siphons,  and  to  insure  their  suc- 
cessful action  pumps  must  be  attached  near  the  highest  eleva- 
tions, which  may  be  occasionally  operated  to  remove  the  air 
that  has  accumulated,  and  which  would  otherwise  cause  the 
flow  to  diminish  and  ultimately  to  cease. 

The  pressure-head,  or  piezometer  height  hl ,  at  any  point  of 
the  pipe  can  be  computed  if  the  velocity  of  flow  is  known,  as 
also  the  depth  H  of  that  point  below  the  water  surface  in  the 
reservoir.  In  the  above  figures  the  ordinate  AlDl  is  the  depth 
H.  Then 


in  which  v  must  be  known  by  measurement  or  be  computed  by 
the  method  of  Art.  76  from  the  total  length  /  and  the  given 
head  h.  This  may  be  put  into  a  simpler  form  by  substituting 
for  v  its  value  in  terms  of  /  and  h,  which  gives 


* 


or  for  long  pipes,  where  \-\-m  may  be  neglected, 

k>  =  H-l-jh  .......    ....    (63)' 

This  formula,  indeed,  can  be  directly  derived  from  the  above 
figures  by  similar  triangles,  taking  the  point  B  as  coincident 
with  A,  which  for  long  pipes  is  allowable,  since  AB  is  very 
small  (Art.  80). 


ART.  85.] 


A   PIPE    WITH  A   NOZZLE. 


193 


The  above  discussion  shows  that  it  is  immaterial  where  the 
pipe  enters  the  reservoir,  provided  that  it  enters  below  the 
hydraulic  gradient  point  B.  It  is  also  not  to  be  forgotten  that 
the  whole  investigation  rests  on  the  assumption  that  the  lengths 
/,  and  /  are  sensibly  equal  to  their  horizontal  projections. 

Prob.  107.  A  pipe  3  inches  in  diameter  discharges  538  cubic 
feet  per  hour  under  a  head  of  12  feet.  At  a  distance  of  300 
feet  from  the  reservoir  the  depth  of  the  pipe  below  the  water 
surface  in  the  reservoir  is  4.5  feet.  Compute  the  probable 
pressure-head  at  this  point.  Ans.  —  0.2  feet. 

ARTICLE  85.  A  PIPE  WITH  A  NOZZLE. 

Water  is  often  delivered  through  a  nozzle  in  order  to  per- 
form work  upon  a  motor  or  for  the  purposes  of  hydraulic  min- 
ing, the  nozzle  being  attached  . 
to  the  end  of  a  pipe  which 
brings  the  flow  from  a  reser- 
voir. In  such  a  case  it  is  de- 
sirable that  the  pressure  at  the 
entrance  to  the  nozzle  should 
be  as  great  as  possible,  and 
this  will  be  effected  when  the  loss  of  head  in  the  pipe  is  as 
small  as  possible.  The  pressure  column  in  a  piezometer,  sup- 
posed to  be  inserted  at  the  end  of  the  pipe,  as  shown  at  ClDl 
in  Fig.  59,  measures  the  pressure-head  there  acting,  and  the 
height  AlCl  measures  the  lost  head  plus  the  velocity-head,  the 
latter  being  very  small. 

Let  h  be  the  total  head  on  the  end  of  the  nozzle,  ^  its 
length,  dl  its  diameter,  and  v^  the  velocity  of  discharge  at  the 
small  end.  Let  /,  d,  and  v  be  the  corresponding  quantities  for 
the  pipe.  Then  the  effective  velocity-head  of  the  issuing  stream 

is  —  ,  and  the  lost  head  is  h — .     This  lost  head  consists  of 


IQ4  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

several  parts  —  that  lost  at  the  entrance  Dl  ;  that  lost  in  friction 
in  the  pipe  ;  that  lost  in  curves  and  valves,  if  any  ;  and  lastly,  that 
lost  in  the  nozzle.  Thus, 

v?  v*  /  v*  v*  v* 

h  --  —  =  m  --  \-f-j  --  h  n  --  h  MI  —  • 

2g  I2g 


Here  m  is  determined  by  Art.  73,  /by  Art.  74,  n  by  Art.  75, 
and  ml  is  to  be  found  from  the  coefficient  of  velocity  of  the 
nozzle  (Art.  63)  in  the  same  manner  as  m.  If,  for  instance,  cl 
for  the  nozzle  is  0.98,  then 


and  for  a  perfect  nozzle  m^  would  be  zero.  The  value  of  m^ 
includes  all  losses  of  head  in  the  nozzle,  as  m  does  in  the  en- 
trance tube,  so  that  the  length  ^  need  not  be  considered. 

The  velocities  v  and  vl  are  inversely  as  the  areas  of  the  cor- 
responding sections,  whence 


Inserting  this  in  the  above  expression,  and  solving  for  z>,  gives 
the  formula 


from  which  v  can  be  computed  by  the  tentative  method  ex- 
plained in  Art.  76.  This  equation,  in  connection  with  the  pre- 
ceding, shows  that  the  greatest  velocity  v^  obtains  when  d  is  as 
large  as  possible  compared  to  dl .  As  the  object  of  a  nozzle  is 
to  utilize  either  the  velocity  or  the  energy  of  the  water,  a  large 
pipe  and  a  small  nozzle  should  hence  be  employed  to  gtve  the 
best  result,  and  this  is  attained  when  the  velocity  vl  is  nearly 
equal  to 


ART.  85.]  A    PIPE    WITH  A   NOZZLE.  195 

As  a  numerical  example,  the  effect  of  attaching  a  nozzle  to 
the  pipe  whose  discharge  was  computed  in  Art.  77  will  be  con- 
sidered. There/—  500,  ^  —  0.25,  and  h  =  4  feet;  7^=0.5, 
n  =  o,  77  =  2.15  feet;  and  q  =  0.106  cubic  feet,  per  second. 
Now  let  the  nozzle  be  one  inch  in  diameter  at  the  small  end,  or 
d^  =•  0.0833  feet  and  cl  =  0.98,  whence  m1  =  0.041.  Using 
/=  0.029,  the  velocity  in  the  pipe  is 


•-*/- 

V  o., 


2  X  32.16  X  4 


,5+ 0.029  X  500  X  4+  1-041  X  81  ' 

whence  77=1.35  feet  per  second.  The  effect  of  the  nozzle, 
therefore,  is  to  reduce  the  velocity,  owing  to  the  loss  of  head 
which  it  causes.  The  velocity  of  flow  from  the  nozzle  is 

z>!  =  1.35  X  9  =  12.15  feet  per  second  ; 
and  the  discharge  per  second  is 

q  =  0.7854  X  0.253  X  1.35  =  0.066  cubic  feet , 

which  is  about  40  per  cent  less  than  that  of  the  pipe  before  the 
nozzle  was  attached.  The  nozzle,  however,  produces  a  marvel- 
lous effect  in  increasing  the  energy  of  the  discharge ;  for  the 
velocity-head  corresponding  to  2.15  feet  per  second  is  only 
0.072  feet,  while  that  corresponding  to  12.15  feet  per  second  is 
2.30  feet,  or  about  32  times  as  great.  As  the  total  head  is  4 
feet,  the  efficiency  of  the  stream  issuing  from  the  nozzle  is 
about  57  per  cent. 

If  the  pressure-head  hl  at  the  entrance  of  the  nozzle  be 
observed,  either  by  a  piezometer  or  by  a  pressure  gauge,  the 
velocity  of  discharge  can  be  computed  by  the  formula 


/ 

V 


whose  demonstration  is  given  in  Art.  63.     If  both  h^  and  vl  be 


196 


FLOW   THROUGH  PIPES. 


[CHAP.  VII. 


measured,  this  formula  furnishes  the  means  of  computing  mt , 
or  the  loss  of  head  caused  by  the  nozzle.* 

Prob.  108.  Compute  the  velocity  and  effective  velocity-head 
for  the  above  pipe  and  nozzle,  but  taking  the  head  h  as  16  feet. 

ARTICLE  86.  HOUSE  SERVICE  PIPES. 

A  service  pipe  which  runs  from  a  street  main  to  a  house  is 
connected  to  the  former  at  right  angles,  and  usually  by  a 
"ferrule"  which  is  'smaller  in  diameter  than  the  pipe  itself. 

The  loss  of  head  at  entrance 
is  hence  larger  than  in  the 
cases  before  discussed,  and  m 
should  probably  be  taken  as 
at  least  equal  to  unity.  The 
pipe,  if  of  lead,  is  frequently 
carried  around  sharp  corners 
by  curves  of  small  radius ;  if 
of  iron,  these  curves  are  formed  by  pieces  forming  a  quadrant 
of  a  circle  into  which  the  straight  parts  are  screwed,  the  radius 
of  the  centre  line  of  the  curve  being  but  little  larger  than  the 
radius  of  the  pipe,  so  that  each  curve  causes  a  loss  of  head 
equal  nearly  to  double  the  velocity-head  (Art.  75).  For  new 
clean  pipes  the  loss  of  head  due  to  friction  may  be  estimated 
by  the  rules  of  Art.  74. 

A  water  main  should  be  so  designed  that  a  certain  minimum 
pressure-head  hv  exists  in  it  at  times  of  heaviest  draught.  This 
pressure-head  may  be  represented  by  the  height  of  the  piezom- 
eter column  AB,  which  would  rise  in  a  tube  supposed  to  be 
inserted  in  the  main,  as  in  Fig.  60.  The  head  h  which  causes 
the  flow  in  the  pipe  is  then  the  difference  in  level  between 

*  For  experimental  formulas  for  flow  through  hose  and  nozzles,  see  a  paper 
by  WESTON  in  Transactions  American  Society  Civil  Engineers,  1884,  vol.  xiii. 
p.  376. 


FIG.  60. 


ART.  86.]  HOUSE   SERVICE  PIPES.  197 

the  top  of  this  column  and  the  end  of  the  pipe,  or  AC.  In- 
serting for  h  this  value,  the  formulas  of  Arts.  76  and  77  may  be 
applied  to  the  investigation  of  service  pipes,  in  the  manner 
there  illustrated.  As  the  sizes  of  common  house-service  pipes 
are  regulated  by  the  practice  of  the  plumbers  and  by  the  market 
sizes  obtainable,  it  is  not  often  necessary  to  make  computations 
regarding  them. 

The  velocity  of  flow  in  the  main  has  no  direct  influence  upon 
that  in  the  pipe,  since  the  connection  is  made  at  right  angles. 
But  as  that  velocity  varies,  owing  to  the  varying  draught  upon 
the  main,  the  pressure-head  /^  is  subject  to  continual  fluctua- 
tions. When  there  is  no  flow  in  the  main,  the  piezometer 
column  rises  until  its  top  is  on  the  same  level  as  the  surface  of 
the  reservoir ;  in  times  of  great  draught  it  may  sink  below  C,  so 
that  no  water  can  be  drawn  from  the  service  pipe. 

The  detection  and  prevention  of  the  waste  of  water  by  con- 
sumers is  a  matter  of  importance  in  cities  where  the  supply 
is  limited  and  where  meters  are  not  in  use.  Of  the  many 
methods  devised  to  detect  this  waste,  one  by  the  use  of  pie- 
zometers may  be  noticed,  by  which  an  inspector  without  enter- 
ing a  house  may  ascertain  whether  water  is  being  drawn  within, 
and  the  approximate  amount  per  second.  Let  M  be  the  street 
main  from  which  a  service  pipe  MOH  runs  to  a  house  H.  At 
the  edge  of  the  sidewalk  a  tube  OP  is  connected  to  the  service 
pipe,  which  has  a  three-way  cock  at  O,  which 
can  be  turned  from  above.  The  inspector, 
passing  on  his  rounds  in  the  night-time,  at- 
taches a  pressure  gauge  at  P  and  turns  the 
cock  O  so  as  to  shut  off  the  water  from  the 
house  and  allow  the  full  pressure  of  the  main  FlG- 6r- 

pl  to  be  registered.  Then  he  turns  the  cock  so  that  the  water 
may  flow  into  the  house,  while  it  also  rises  in  OP  and  registers 
the  pressure  /, .  Then  if  /a  is  less  than  pl  it  is  certain  that  a 


198  FLO IV   THROUGH  PIPES.  [CHAP.  VII. 

waste  is  occurring  within  the  house,  and  the  amount  of  this  may 
be  approximately  computed  if  desired,  in  the  manner  indicated 
in  Art.  70,  and  the  consumer  be  fined  accordingly.* 

Prob.  109.  Describe  a  water-pressure  regulator  to  be  placed 
between  the  main  and  the  house  so  that  the  pressure  in  the 
service  pipes  may  never  exceed  a  given  quantity — say  40  pounds 
per  square  inch. 

ARTICLE  87.  A  WATER  MAIN. 

The  simplest  case  of  the  distribution  of  water  is  that  where 
a  single  main  is  tapped  by  a  number  of  service  pipes  near  its 
end,  as  shown  in  Fig.  62.  In  designing  such  a  main  the  prin- 
cipal consideration  is  that  it  should 


*"T~ 

•*  E5      be  large  enough  so  that  the  pres- 

sure-head /ilf  when  all  the  pipes 
are  in  draught,  shall  be  amply  suf- 
ficient to  deliver  the  water  into 
the  highest  houses  along  the  line. 


FIG.  62.  FANNING    recommends    that   this 

pressure-head  in  commercial  and  manufacturing  districts  should 
not  be  less  than  150  feet,  and  in  suburban  districts  not  less 
than  100  feet.  The  height  H  to  the  surface  of  the  water  in 
the  reservoir  will  always  be  greater  than  h^ ,  and  the  pipe  is  to 
be  so  designed  that  the  losses  of  head  may  not  reduce  //x 
below  the  limit  assigned.  The  head  h  to  be  used  in  the  for- 
mulas is  the  difference  H  —  hl .  The  discharge  per  second  q 
being  known  or  assumed,  the  problem  is  to  determine  the 
diameter  d  of  the  main. 

A  strict  theoretical  solution  of  even  this  simple  case  leads 
to  very  complicated  calculations,  and  in  fact  cannot  be  made 
without  knowing  all  the  circumstances  regarding  each  of  the 
service  pipes.  Considering  that  the  result  of  the  computation 

*  This  briefly  describes  CHURCH'S  water-waste  indicator. 


ART.  87.]  A    WA  TER  MAIN.  199 

is  merely  to  enable  one  of  the  market  sizes  to  be  selected,  it  is 
plain  that  great  precision  cannot  be  expected,  and  that  ap- 
proximate methods  may  be  used  to  give  a  solution  entirely 
satisfactory.  It  will  then  be  assumed  that  the  service  pipes 
are  connected  with  the  main  at  equal  intervals,  and  that  the 
discharge  through  each  is  the  same  under  maximum  draught. 
The  velocity  v  in  the  main  then  decreases,  and  becomes  o  at 
the  dead  end.  The  loss  of  head  per  linear  foot  in  the  length 
4  (Fig.  62)  is  hence  less  than  in  /.  To  estimate  this,  let  vl  be 
the  velocity  at  a  distance  x  from  the  dead  end ;  then 


The  loss  of  head  in  friction  in  the  length  dx  is 
*///       *Sx  v?       s  **    V*  * 

=  f^^g=f-dr?TgSx' 

and  hence  between  the  limits  o  and  /,  that  loss  is 

^'=/M'  • (6s) 

provided  that  f  remains  constant.  This  is  really  not  the  case, 
but  no  material  error  is  thus  introduced,  since/"  must  be  taken 
larger  than  the  tabular  values  in  order  to  allow  for  the  deteri- 
oration of  the  inner  surface  of  the  main.  The  loss  of  head  in 
friction  for  a  pipe  which  discharges  uniformly  along  its  length 
may  therefore  be  taken  at  one-third  of  that  which  occurs  when 
the  discharge  is  entirely  at  the  end. 

Now  neglecting  the  loss  of  head  at  entrance  and  the  effec- 
tive velocity-head  of  the  discharge,  the  total  head  h  is  entirely 
consumed  in  friction,  or 


20O  FLOW   THROUGH  PIPES.  [CHAP.  VIL 

Placing  in  this  for  v  its  value  in  terms  of  the  total  discharge  q, 
and  solving  for  d,  gives 


This  is  the  same  as  the  formula  of  Art.  80,  except  that  /  has 
been  replaced  by  /  -f-  J/x  .     The  diameter  in  feet  then  is 


as  in  the  case  of  long  pipes. 

For  example,  consider  a  village  consisting  of  a  single  street, 
whose  length  l^  —  3000  feet,  and  upon  which  there  are  100 
houses,  each  furnished  with  a  service  pipe.  The  probable 
population  is  then  500,  and  taking  100  gallons  per  day  as  the 
consumption  per  capita,  this  gives  the  average  discharge  per 

second 

500  X  ioo 

a  =  -  ~-   ~2  --  =  0-°774  cubic  feet  ; 
7.48  X  36oo  X  24 

and  as  the  maximum  draught  is  often  double  of  the  average, 
q  will  be  taken  as  0.15  cubic  feet  per  second.  The  length  / 
to  the  reservoir  is  4290  feet,  whose  surface  is  90.5'  feet  above 
the  dead  end  of  the  main,  and  it  is  required  that  under  full 
draught  the  pressure-head  in  the  main  shall  be  75  feet.  Then 
h  =  90.5  —  75  =  15.5  feet,  and  taking/  =  0.03  in  order  to  be 
on  the  safe  side,  the  formula  gives 

d  =  0.36  feet  =  4.3  inches. 

Accordingly  a  four-inch  pipe  is  nearly  large  enough  to  satisfy 
the  imposed  conditions. 

To  consider  the  effect  of  fire  service  upon  the  diameter  of 
the  main,  let  there  be  four  hydrants  placed  at  equal  intervals 
along  the  line  /x  ,  each  of  which  is  required  to  deliver  20  cubic 
feet  per  minute  under  the  same  pressure-head  of  75  feet.  This 
gives  a  discharge  1.33  cubic  feet  per  second,  or,  in  total, 


ART.  88.] 


A   MAIN    WITH  BRANCHES. 


2O I 


q  =  1.33  -\-  0.15  =  1.5  cubic  feet.     Inserting  this  in  the  for. 
mula,  and  using  for/" the  same  value  as  before, 
d  =  0.897  feet  =  10.8  inches. 

Hence  a  ten-inch  pipe  is  at  least  required  to  maintain  the 
required  pressure  when  the  four  hydrants  are  in  full  draught  at 
the  same  time  with  the  service  pipes. 

Prob.  1 10.  Compute  the  velocity  v  and  the  pressure-head  hl 
for  the  above  example,  if  the  main  be  10  inches  in  diameter 
and  the  discharge  1.5  cubic  feet  per  second. 

ARTICLE  88.  A  MAIN  WITH  BRANCHES. 
In  Fig.  63  is  shown  a  main  of  length .  /  and  diameter  d, 
having  two  branches  with  lengths  /x  and  /3 ,  and  diameters  */, 

T" 
w, 


FIG.  63. 

and  d^  .  These  being  given,  as  also  the  heads  H^  and  H9  under 
which  the  flow  occurs,  it  is  required  to  find  the  discharges  ql  and 
q^  .  Let  v,  vl  ,  and  v^  be  the  corresponding  velocities  ;  then  for 
long  pipes,  in  which  all  losses  except  those  due  to  friction  may 
be  neglected,  a 


where  y  is  the  difference  in  level  between  the  reservoir  surface 
and  the  water  level  in  a  piezometer  supposed  to  be  inserted  at 
the  junction.  Thisjj/  is  the  friction-head  consumed  in  the  large 


main,  or 


/    V* 


2O2      •  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

Inserting  this  in  the  two  equations,  and  placing  for  the  veloci- 
ties their  values  in  terms  of  the  discharges,  they  become 


which  q^  and  q^  are  best  obtained  by  trial  ;  although  by 
solution  the  value  of  each  may  be  directly  expressed  in  terms 
of  the  given  data,  the  expressions  are  too  complicated  for 
general  use. 

When  it  is  required  to  determine  the  diameters  from  the 
given  lengths,  heads,  and  discharges,  there  are  three  unknown 
quantities,  d,  dl  ,  d^  ,  to  be  found  from  only  two  equations,  and 
the  problem  is  indeterminate.  If,  however,  d  be  assumed, 
values  of  dl  and  </2  may  be  found  ;  and  as  d  may  be  taken  at 
pleasure,  it  appears  that  an  infinite  number  of  solutions  is  pos- 
sible. Another  way  is  to  assume  a  value  of  y,  corresponding 
to  a  proper  pressure-head  at  the  junction  ;  then  the  diameters 
are  directly  found  from  the  usual  formula  for  long  pipes, 


in  which  h  is  replaced  by  y  for  the  large  main,  and  by  Hl  —  y 
and  H9  —  y  for  the  smaller  ones,  q  for  the  first  being  q^  -\-  q^ , 
and  for  the  others  ql  and  qz  respectively. 

A  water-supply  system  consists  of  a  principal  main  with  many 
sub-mains  as  branches.  In  designing  these  the  quantities  of 
water  to  be  furnished  are  assumed  from  the  present  and  prob- 
able future  population,  which  in  small  towns  requires  from  40 
to  100  gallons  per  capita  per  day,  and  in  large  cities  from  100 
to  1 50  gallons.  This  should  be  furnished  under  heads  sufficient 
to  raise  the  water  into  the  highest  houses,  as  also  for  use  in 


ART.  89.]  PUMPING    THROUGH  PIPES.  2O$ 

cases  of  fire.  As  the  problem  of  computing  the  diameters 
from  the  given  data  is  indeterminate,  it  will  probably  be  as  well 
to  assume  at  the  outset  the  sizes  for  the  principal  mains.  The 
velocities  corresponding  to  the  given  quantities  and  the  as- 
sumed sizes  are  then  computed,  and  from  these  the  pressure- 
heads  at  a  number  of  points  are  found.  If  these  are  not  satis- 
factory, other  sizes  are  to  be  taken  and  the  computation  be 
repeated.  The  successful  design  will  be  that  which  will  furnish 
the  required  quantities  under  proper  pressures  with  the  least 
expenditure. 

Prob.  ill.  In  Fig.  63  let  q^  —  0.5  and  q^  =  0.4  cubic  feet ; 
Hl  —  140  and  H^  —  125  feet;  /x  =  3810,  ^=  2455,  and  /  = 
12  314  feet.  If  d^  equals  d^  find  the  values  of  d  and  dl9  and 
also  the  pressure-head  at  the  junction  if  its  depth  below  the 
reservoir  level  is  108  feet. 


ARTICLE  89.  PUMPING  THROUGH  PIPES. 

When  water  is  pumped  through  a  pipe  from  a  lower  to  a 
higher  level,  the  power  of  the  pump  must  be  sufficient  not  only 
to  raise  the  required  amount  in  a  given  time,  but  also  to  over- 
come the  various  resistances  to  flow.  The  head  due  to  the  re- 
sistances is  thus  a  direct  source  of  loss,  and  it  is  desirable  that 
the  pipe  be  so  arranged  as  to  render  this  as  small  as  possible. 

Let  w  be  the  weight  of  a  cubic  foot  of  water  and  q  the 
quantity  raised  per  second  through  the  height  //",  which,  for 
example,  may  be  the  difference  in  level  t  i 

I       •=-- ^1  T 

between  a  canal  C  and  a  reservoir  R,  as 
in  Fig.  64.  The  useful  work  done  by 
the  pump  in  each  second  is  wqH.  Let  h' 
be  the  head  lost  in  entering  the  pipe  at 
the  canal,  h"  that  lost  in  friction  in  the 
pipe,  and  h'"  all  other  losses  of  head,  such  as  those  caused  by 


2O4  FLOW   THROUGH  PIPES.  [CHAP.  VII. 

curves,  valves,  and  by  resistances  in  passing  through  the  pump 
cylinders.  Then  the  total  work  performed  by  the  pump  per 
second  is 

q(k'  +  h"  +  h'").     .     .     .      (67) 


Inserting  for  the  lost  heads  their  values,  this  becomes 

l+n.     .     .    .    (67)' 


In  order,  therefore,  that  the  losses  of  work  may  be  as  small  as 
possible,  the  velocity  of  flow  through  the  pipe  should  be  low  ; 
and  this  is  to  be  effected  by  making  the  diameter  of  the  pipe 
large. 

For  example,  let  it  be  required  to  determine  the  horse- 
power of  a  pump  to  raise  I  200  ooo  gallons  per  day  through  a 
height  of  230  feet,  when  the  diameter  of  the  pipe  is  6  inches 
and  its  length  1400  feet.  The  discharge  per  second  is 

i  200000  0/;      ,  .    , 

q  =  -  -  —  =  1.86  cubic  feet. 
7.481  X  24  X  3600 

and  the  velocity  of  flow  is 

1.86 

v  =  —  5—  «  —  -  -  5  =  9.47  feet  per  second. 
0.7854  X  o.S 

The  probable  head  lost  at  entrance  into  the  pipe  is 
K  =  0.5—  ==  0.5  X  I«39  =  0.7  feet. 

o 

When  the  pipe  is  new  and  clean  the  friction  factor  f  is 
about  C.O2O,  as  shown  by  Table  XVI  ;  then  the  loss  of  head 
in  friction  is 

h"  =  0.020  X  ~~  X  1-39  =  77-8  feet. 


ART.  89.]  PUMPING    THROUGH  PIPES.  2O5 

The  other  losses  of  head  depend  upon  the  details  of  the  valve 
and  pump  cylinder ;  if  these  be  such  that  n  =  4,  then 

//'"  =  4X  i.39=  5-6  feet. 
The  total  losses  of  head  hence  are 

#-[- /*"  +  £'"  =  84. 1  feet. 
The  work  to  be  performed  per  second  by  the  pump  now  is 

k  =  62.5  X  1.86(230  4~  84.1)  =  36  510  foot-pounds, 
and  the  horse-power  expended  is 

^^36510^66.4. 
550 

If  there  were  no  losses  in  friction  and  other  resistances  the 
work  done  would  be  simply 

k  =  62.5  X  1.86  X  230  =  26  740  foot-pounds, 
and  the  corresponding  horse-power  would  be 

lgp=  26740         8A 
550 

Accordingly  17.8  horse-power  is  wasted  in  injurious  resistances. 

i 

For  the  same  data  let  the  6-inch  pipe  be  replaced  by  one 
14  inches  in  diameter.  Then,  proceeding  as  before,  the  velocity 
of  flow  is  found  to  be  1.80  feet  per  second,  the  head  lost  at 
entrance  0.03  feet,  the  head  lost  in  friction  1.23  feet,  and  that 
lost  in  other  ways  0.20  feet.  The  total  losses  of  head  are  thus 
only  1.46  feet,  as  against  84.1  feet  for  the  smaller  pipe,  and  the 
horse-power  required  is  48.9,  which  is  but  little  greater  than 
the  theoretic  power.  The  great  advantage  of  the  larger  pipe 
is  thus  apparent,  and  by  increasing  its  size  to  18  inches  the 
losses  of  head  may  be  reduced  so  low  as  to  be  scarcely  appre- 
ciable in  comparison  with  the  useful  head  of  230  feet. 

A  pump  is  often  used  to  force  water  directly  through  the 


2O6 


FLOW   THROUGH  PIPES. 


[CHAP.  VIL 


mains  of  a  water-supply  system  under  a  designated  pressure. 

The  work  of  the  pump  in  this 
case  consists  of  that  required  to 
maintain  the  pressure  and  that 
required  to  overcome  the  fric- 
tional  resistances.  Let  h^  be  the 
pressure-head  to  be  maintained 
at  the  end  of  the  main,  and  z  the 
height  of  the  main  above  the 
level  of  the  river  from  which  the 


FIG.  65. 


water  is  pumped  ;  then  hl  -f-  z  is  the  head  H,  which  corresponds 
to  the  useful  work  of  the  pump,  and,  as  before, 

k  =  wqH  +  wq(ti  +  h"  +  h'"). 

To  reduce  these  injurious  heads  to  the  smallest  limits  the 
mains  should  be  large  in  order  that  the  velocity  of  flow  may 
be  small.  In  Fig.  65  is  shown  a  symbolic  representation  of 
the  case  of  pumping  into  a  main,  P  being  the  pump,  C  the 
source  of  supply,  and  DM  the  pressure-head  which  is  main- 
tained upon  the  end  of  the  pipe  during  the  flow.  At  the 
pump  the  pressure-head  is  AP,  so  that  AD  represents  the  hy* 
draulic  gradient  for  the  pipe  from  P  to  M.  The  total  work  of 
the  pump  may  then  be  regarded  as  expended  in  lifting  the  water 
from  C  to  Ay  and  this  consists  of  three  parts,  corresponding  to 
the  heads  CM  or  z,  MD  or  hlt  and  AB  or  h1  +  h"  -f  h'",  the 
first  overcoming  the  force  of  gravity,  the  second  delivering 
the  flow  under  the  required  pressure,  while  the  last  is  trans- 
formed into  heat  in  overcoming  friction  and  other  resistances. 
In  this  direct  method  of  water  supply  a  standpipe,^/7,  is  often 
erected  near  the  pump,  in  which  the  water  rises  to  a  height 
corresponding  to  the  required  pressure,  and  which  furnishes 
a  supply  when  a  temporary  stoppage  of  the  pumping  engine 
occurs. 

Prob.  112.  Compute  the  horse-power  of  a  pump  for  the  fol- 


ART.  90.]  LEATHER  AND  RUBBER  HOSE.  2O/ 

lowing  data,  neglecting  all  resistances  except  those  due  to  fric- 
tion: £—  1.5  cubic  feet  per  second,  which  is  distributed  uni- 
formly over  a  length  ^  =  3000  feet,  the  remaining  length  o! 
the  pipe  being  4290  feet  ;  d  =  10  inches,  kl  =  75  feet,  z  =  0.4 
feet. 

ARTICLE  90.  LEATHER  AND  RUBBER  HOSE. 

The  losses  of  head  in  friction  are  greater  in  leather  and 
rubber  hose  than  in  clean  iron  pipes,  especially  so  at  low  ve- 
locities. The  following  are  values  of  the  friction  factor  f 
which  have  been  deduced  from  experiments  made  by  ELLIS,** 
on  hose  2%  inches  in  diameter,  to  which  are  added  the  values 
for  an  iron  pipe  of  the  same  size  : 

For  velocity  v        =          3           4           6          10         15  25  feet, 

ffor  leather  hose  =  0.095  0.064  0.043  0.033  0.030  0.029 

f  for  rubber  hose  =  0.045  0.033  0.027  0.025  0.026  0.027 

fior  iron  pipe        =  0.027  0.026  0.025  0.023  0.022 

By  the  help  of  this  table  computations  may  be  made  on  the 
pumping  of  water  through  hose  for  delivery  in  fire  streams  or 
for  other  purposes,  in  the  same  manner  as  for  pipes. 

The  loss  of  head  in  a  long  hose  becomes  so  great  even 
under  moderate  velocities  as  to  consume  a  large  proportion  of 
the  pressure  exerted  by  the  hydrant  or  steamer.  For  example, 
let  this  primitive  pressure  be  122  pounds  per  square  inch,  cor- 
responding to  a  head  of  281  feet,  and  let  it  be  required  to  find 
the  pressure-head  in  the  2^-inch  leather  hose  at  1000  feet  dis- 
tance, when  a  nozzle  is  used,  which  discharges  153  gallons  per 
minute,  the  hose  being  laid  horizontal.  In  cubic  feet  per 
second  the  discharge  is 


G.  A.  ELLIS,  Fire  Streams  (Springfield,  1878). 


208  FLOW   THROUGH  PIPES.  [CHAP.  VII, 

and  the  velocity  in  the  hose  is  accordingly  found  to  be 

q 
v  =  T~~ji  —  IO-°  ^eet  Per  second 

Hence  the  loss  of  head  in  friction  is 


and    consequently  the  pressure-head   at   the  entrance  to  the 

nozzle  is 

h^  =  281  —  246  =  35  feet, 

which  corresponds  to  about  15  pounds  per  square  inch.  The 
remedy  for  this  great  reduction  of  pressure  is  to  employ  a 
smaller  nozzle,  thus  decreasing  the  discharge  and  the  velocity 
in  the  hose  ;  but  if  both  head  and  quantity  of  discharge  are 
desired  they  can  only  be  secured  either  by  an  increase  of  pres- 
sure at  the  steamer  or  by  the  use  of  a  larger  hose. 

Prob.  113.  When  the  pressure  gauge  at  the  steamer  indi- 
cates 83  pounds  per  square  inch,  a  gauge  on  the  leather  hose 
800  feet  distant  reads  25  pounds.  Compute  the  value  of  the 
friction  factor/",  the  discharge  per  minute  being  121  gallons. 

Ans.  0.036. 

ART.  91.  LAMPE'S  FORMULA. 

There  have  been  made  many  attempts  to  express  the  mean 
velocity  v  without  the  use  of  a  factor  or  coefficient  of  friction. 
That  this  can  be  empirically  done,  within  the  range  of  experi- 
mental results,  is  plain  by  observing  that  the  values  of  f  in 
Table  XVI  show  a  regular  variation  with  the  diameter  d.  For 
long  pipes  /is  then  a  function  of  d  and  v,  or  a  function  of  d,  h, 
and  /.  The  simplest  expression  of  the  relation  between  these 
quantities  is 


ART.  9  1.]  LAMPE'S  FORMULA.  209 

in  which  a,  fi,  and  y  are  empirical  constants.  The  investiga- 
tions of  LAMPE  have  determined  probable  values  for  these 
constants,  giving 

0-555 

,  ......  (eg) 

in  which  d,  h,  and  /  are  to  be  taken  in  feet,  and  v  will  be 
found  in  feet  per  second,  This  formula  is  only  applicable  to 
long  circular  pipes  with  surfaces  clean  or  in  fair  condition. 

From  this  formula  the  discharge  q  may  be  expressed 

(68)' 

and  the  diameter  required  to  discharge  a  given  quantity  is 

0-206 


By  the  use  of  these  formulas  all  of  the  preceding  problems 
-concerning  long  pipes  may  be  directly  solved  without  the  use 
of  the  tables  of  friction  factors.  They  show  that  the  discharg- 
ing capacity  of  long  pipes  varies  about  as  the  2.7  power  of  the 
diameter  (Art.  80). 

As  an  example,  let  it  be  required  to  find  the  diameter  of  a 
pipe  which  is  to  discharge  177  300  gallons  per  hour,  its  length 
being  75  ooo  feet  and  the  head  135  feet.  Here 


q  =  --—-TT  =  6.583  cubic  feet. 
3600  X  748i 

/      75  ooo 

and  -r——  ---  =  555.6; 

h         135 

whence  by  the  formula 

^= 
which  gives 

a?  =  -i.6  1  feet  —  19.3  inches, 

so  that  a  2O-inch  pipe  should  be  selected. 

Prob.  114.  Solve  Problems  102  and   103  by  the  use  of  the 
above  formulas. 


2IO  FLOW   THROUGH  PIPES.  [CHAP.  VIL 


ARTICLE  92.  VERY  SMALL  PIPES. 

The  preceding  investigations  and  rules  apply  to  pipes  greater 
than  about  0.5  inches  in  diameter,  and  are  not  valid  for  v,ery 
small  pipes.  The  laws  of  discharge  in  these  are  not  understood 
from  a  theoretical  basis,  but  experiments  made  by  POISEUILLE 
in  1843,  m  order  to  study  the  phenomena  of  the  flow  of  blood 
in  veins  and  arteries,  have  settled  beyond  question  that  they 
are  materially  different  from  those  which  govern  large  pipes  at 
ordinary  velocities.  His  investigations  proved  that  for  pipes 
whose  diameters  are  less  than  about  0.7  millimeters  or  0.03 
inches,  the  velocity  is  expressed  by  the  simple  relation 

hd* 
v  —  a—, 

in  which  a  is  a  factor  nearly  constant  at  a  given  temperature. 
The  velocity  then  varies  directly  with  the  head  and  with  the 
square  of  the  diameter,  and  inversely  with  the  length.  It  is 
here  supposed  that  the  pipe  is  long,  so  that  losses  of  head  due 
to  entrance  may  be  neglected. 

Later  researches  indicate  that  these  laws  are  also  true  for 
large  pipes,  provided  the  velocity  be  small ;  and  that  for  a 
given  pipe  there  is  a  certain  critical  velocity  at  which  the  law 
changes,  and  beyond  which 


I 

as  for  the  case  of  common  pipes.  This  critical  point  appears 
to  be  that  at  which  the  filaments  cease  to  move  in  parallel 
lines,  and  pass  in  sinuous  paths  from  one  side  of  the  pipe  to- 
the  other.  For  a  very  small  pipe  the  velocity  may  be  high 
before  this  point  is  reached  ;  for  a  large  pipe  it  happens  at  very 
low  velocities. 


ART.  92.]  VERY  SMALL  PIPES.  211 

In  Art.  74  it  was  mentioned  that  the  frictional  resistances 
in  a  pipe  consist  of  those  along  the  inner  surface,  and  of  those 
met  among  the  particles  in  their  sinuous  motion.  Since  in 
small  pipes  the  latter  do  not  exist,  it  appears  from  PoiSEUlLLE's 
formula  that  the  head  lost  in  friction  along  the  inner  surface 
may  be  expressed  by 

*"^4- 

ad* 

Now  if  the  law  were  known  which  governs  the  loss  in  internal 
friction  it  might  be  possible  to  add  this  to  the  preceding,  and 
thus  obtain  an  expression  for  loss  of  head  in  which  the  friction 
factor  would  be  a  quantity  dependent  only  upon  the  nature  of 
the  surface.  Thus  far,  however,  efforts  in  this  direction  have 
not  been  practically  successful. 

The  effect  of  temperature  on  the  flow  has  not  been  consid- 
ered in  the  previous  articles,  and  in  fact  but  little  is  known  re- 
garding it,  except  that  a  very  slight  increase  in  discharge  is  prob- 
able for  a  high  rise  in  the  temperature  of  the  water.  For  very 
small  pipes,  however,  PoiSEUILLE  found  that  a  marked  in- 
crease in  velocity  and  discharge  resulted,  the  value  of  a  being 
about  twice  as  great  at  45°  Centigrade  as  at  10°. 

Prob.  115.  The  value  of  a  for  small  pipes  is  about  184 
when  h,  d,  /,  and  v  are  in  millimeters,  and  the  flow  occurs  at 
10°  Centigrade.  Find  its  value  when  the  foot  is  the  unit  of 
measure. 


212  FLO IV  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 


CHAPTER   VIII. 
FLOW  IN  CONDUITS  AND  CANALS. 

ARTICLE  93.  DEFINITIONS. 

Water  is  often  conveyed  from  place  to  place  in  artificial 
channels,  such  as  troughs,  aqueducts,  ditches,  and  canals,  there 
being  no  head  to  cause  the  flow  except  that  due  to  the  slope. 
The  word  conduit  will  be  used  as  a  general  term  for  a  channel 
lined  with  timber,  mortar,  or  masonry,  and  will  also  include 
metal  pipes,  troughs,  and  sewers.  Conduits  may  be  either 
open  as  in  the  case  of  troughs,  or  closed  as  in  sewers  and  most 
aqueducts.  Streams  flow  in  natural  channels  eroded  in  the 
earth,  and  include  small  brooks  as  well  as  the  largest  rivers. 
Most  of  the  principles  relating  to  conduits  and  canals  apply 
also  to  streams,  and  the  word  channel  will  be  used  as  applica- 
ble to  all  classes. 

The  wetted  perimeter  of  the  cross-section  of  a  channel  is 
that  part  of  its  boundary  which  is  in  contact  with  the  water. 
Thus,  if  a  circular  sewer  of  diameter  d  be  half  full  of  water 
the  wetted  perimeter  is  \nd.  In  this  chapter  the  letter  p  will 
designate  the  wetted  perimeter. 

The  hydraulic  radius  of  a  water  cross-section  is  its  area 
divided  by  its  wetted  perimeter.  Let  a  be  the  area  and  r  the 
hydraulic  radius  ;  then 

a 

7  —  — . 

P 

The  letter  r  is  of  frequent  occurrence  in  formulas  for  the  flow 


ART.  93.]  DEFINITIONS.  213 

in  channels  ;  it  is  a  linear  quantity,  which  is  always  expressed 
in  the  same  unit  as  /.     It 


or  hy- 

draulic   mean    depth,    be-  FlG-  66' 

cause  for  a  shallow  section  its  value  is  but  little  less  than  the 
mean  depth  of  the  water.  Thus  in  Fig.  66,  if  b  be  the  breadth 
on  the  water  surface,  the  mean  depth  is  a  -f-  b,  and  the  hy- 
draulic radius  is  a  -r-  p]  and  these  are  nearly  equal,  since  p  is 
but  slightly  larger  than  b. 

The  hydraulic  radius  of  a  circular  cross-section  filled  with 
water  is  one-fourth  of  the  diameter  ;  thus  : 


The  same  value  is  also  applicable  to  a  circular  section  half 
filled  with  water,  since  then  both  area  and  wetted  perimeter 
are  one-half  their  former  values. 

The  slope  of  the  water  surface  in  the  longitudinal  section, 
designated  by  the  letter  s,  is  the  ratio  of  the  fall  h  to  the  length 
/  in  which  that  fall  occurs,  or 

s  =  ~ 

The  slope  is  hence  expressed  as  an  abstract  number,  which  is 
independent  of  the  system  of  measures  employed.  To  deter- 
mine its  value  with  precision  h  must  be  obtained  by  referring 
the  water  level  at  each  end  of  the  line  to  a  bench  mark  by  the 
help  of  a  hook  gauge  or  other  accurate  means,  the  benches  be- 
ing connected  by  level  lines  run  with  care.  The  distance  /  is 
measured  along  the  inclined  channel,  and  it  should  be  of  con- 
siderable length  in  order  that  the  relative  error  in  h  may  not 
be  large. 


214  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

If  there  be  no  slope,  or  s  =  o,  there  can  be  no  flow.  But  if 
there  be  even  the  smallest  slope  the  force  of  gravity  furnishes 
a  component  acting  down  the  inclined  surface,  and  motion  en- 
sues. The  velocity  of  flow  evidently  increases  with  the  slope. 

The  flow  in  a  channel  is  said  to  be  permanent  when  the 
same  quantity  of  water  per  second  passes  through  each  cross- 
section.  If  an  empty  channel  be  filled  by  admitting  water  at 
its  upper  end  the  flow  is  at  first  non-permanent  or  variable,  for 
more  water  passes  through  one  of  the  upper  sections  per  second 
than  is  delivered  at  the  lower  end.  But  after  sufficient  time 
has  elapsed  the  flow  becomes  permanent  ;  when  this  occurs  the 
mean  velocities  in  different  sections  are  inversely  as  their 
areas  (Art.  19). 

Uniform  flow  is  that  particular  case  of  permanent  flow 
where  all  the  water  cross-sections  are  equal,  and  the  slope  of  the 
water  surface  is  parallel  to  that  of  the  bed  of  the  channel.  If 
the  sections  vary  the  flow  is  said  to  be  non-uniform,  or  variable, 
although  the  condition  of  permanency  is  still  fulfilled.  In  this 
chapter  only  the  case  of  uniform  flow  will  be  discussed. 

The  velocities  of  different  filaments  in  a  channel  are  not 
equal,  as  those  near  the  wetted  perimeter  move  slower  than 
the  central  ones  owing  to  the  retarding  influence  of  friction. 
The  mean  of  all  the  velocities  of  all  the  filaments  in  a  cross- 
section  is  called  the  mean  velocity  v.  Thus  if  v',  vfl  ',  etc.,  be 
velocities  of  different  filaments, 


(69) 


in  which  n  is  the  number  of  filaments.     Let  a  be  the  area  of 
the  cross-section  and  a'  that  of  one  of  the  elementary  filaments  ; 

then  n  =  —  -,  ,  and  hence 
a 

av  =  a'(v'  +  v"  +  etc.). 


ART.  94.]  FORMULA   FOR  MEAN    VELOCITY.  215. 

But  the  second  member  is  the  discharge  q.     Therefore  the 
mean  velocity  may  be  also  determined  by  the  relation 


v  =  — 
a 


(69)' 


The  filaments  which  are  here  considered  are  in  part  imaginary, 
for  experiments  show  that  there  is  a  constant  sinuous  motion 
of  particles  from  one  side  of  the  channel  to  the  other.  The 
best  definition  for  mean  velocity  hence  is,  that  it  is  a  velocity 
which  multiplied  by  the  area  of  the  cross-section  gives  the  dis- 
charge, or  v  =  q  -i-  a. 

Prob.  116.  Compute  the  hydraulic  radius  of  a  rectangular 
trough  whose  width  is  4.4  feet  and  depth  2.2  feet. 

Prob.  117.  Compute  the  mean  velocity  in  a  circular  sewer 
of  4  feet  diameter  when  it  is  half  filled  and  discharges  120  gal- 
lons per  second. 

ARTICLE  94.  FORMULA  FOR  MEAN  VELOCITY. 

When  all  the  wetted  cross-sections  of  a  channel  are  equal, 
and  the  water  is  neither  rising  nor  falling,  having  attained  a  con- 
dition of  permanency,  the  flow  is  said  to  be  uniform.  This  is 
the  case  in  a  conduit  or  canal  of  constant  size  and  slope  whose 
supply  does  not  vary.  The  same  quantity  of  water  per  second 
then  passes  each  cross-section,  and  consequently  the  mean 
velocity  in  each  section  is  the  same.  This  uniformity  of  flow 
is  due  to  the  resistances  along  the  interior  surface  of  the  chan- 
nel, for  were  it  perfectly  smooth  the  force  of  gravity  would 
cause  the  velocity  to  be  accelerated.  The  entire  energy  of 
the  water  due  to  the  fall  h  is  hence  expended  in  overcoming 
frictional  resistances  along  the  length  /.  Let  W  be  the  weight 
of  water  per  second  which  passes  any  cross-section,  F  the  force 
of  friction  or  resistance  per  square  foot  of  the  interior  surface 
of  the  channel,  /  the  wetted  perimeter,  and  v  the  mean  veloci- 


2l6  FLOW  IN  CONDUITS  AND    CANALS       [CHAP.  VIII. 

ty.  Now  assuming  that  the  friction  is  uniform  over  the  entire 
inner  surface  whose  area  is  //,  the  total  resisting  force  is  Fplr 
and  again  assuming  that  the  velocity  along  the  surface  is  the 
same  as  z/,  the  total  resisting  work  is  Fplv.  Hence 

Fplv  =  Wh. 

But  the  value  of  W  is  wav  where  a  is  the  area  of  the  cross- 
section,  and  w  is  the  weight  of  a  cubic  unit  of  water ;  accord- 
ingly, 

Fpl  =  wah> 
or 

ah 

I*   =  W =•• 

Pi 

a  h 

Here  -  is  the  hydraulic  radius  r,  and  j  is  the  slope  s,  and  the 

value  of  Fis 

F  =  wrs. 

This  is  an  approximate  expression  for  the  resisting  force  of 
friction  per  square  foot  of  the  interior  surface  of  the  channel. 

In  order  to  establish  a  formula  for  mean  velocity  the  value- 
of  F  must  be  expressed  in  terms  of  v,  and  this  can  only  be 
done  by  studying  the  results  of  experiments.  These  indicate 
that  F  is  .approximately  proportional  to  the  square  of  the  mean 
velocity.  Therefore,  if  c  be  a  constant, 

v  =  c  Vrs.     .......      (70) 

This  is  an  empirical  expression  for  the  law  of  variation  of  the 
mean  velocity  with  the  hydraulic  radius  and  slope  of  the  chan- 
nel. The  quantity  c  is  a  coefficient  which  varies  with  the 
degree  of  roughness  of  the  bed  and  with  other  circumstances. 
It  is  the  object  of  the  following  articles  to  state  values  of  c  for 
different  classes  of  conduits  and  canals. 

Another  method  of  establishing  the  above  formula  is  simi- 


ART.  94.]  FORMULA   FOR  MEAN   VELOCITY.  21  J 

lar  to  that  used  in  Art.  74  for  pipes.  The  total  head  h  repre- 
sents the  loss  of  head  in  friction  ;  this  should  vary  directly  with 
/  and  /,  and  it  should  vary  inversely  with  a,  because  for  a  given 
wetted  perimeter  the  friction  will  be  the  least  for  the  largest  a. 
It  should  also  vary  as  the  square  of  the  velocity.  Hence 


a  2g 

in  which/7  is  an  abstract  number  depending  upon  the  charac- 
ter of  the  surface.     From  this  the  value  of  v  is 


-- 


in  which  c  is  the  square  root  of  2f~r~aff.  Notwithstanding 
these  reasonings  the  formula  cannot  be  called  rational;  it  is 
merely  an  empirical  expression  whose  basis  is  experiment. 

To  determine  values  of  the  coefficient  c  the  quantities  v,  ?% 
and  s  are  measured  for  particular  cases,  and  then  c  is  computed. 
To  find  r  and  s  linear  measurements  are  alone  required.  To 
determine  v  the  flow  must  be  gauged  either  in  a  measuring 
vessel  or  by  an  orifice  or  weir,  or,  if  the  channel  be  large,  by 
floats  or  other  indirect  methods  described  in  the  next  chapter. 
It  being  a  matter  of  great  importance  to  establish  a  satisfactory 
formula  for  mean  velocity,  thousands  of  such  gaugings  have 
been  made,  and  from  the  records  of  these  the  values  of  the 
coefficients  have  been  deduced.  It  is  found  that  c  lies  between 
30  and  1  60  when  v  and  r  are  expressed  in  feet,  and  that  its 
value  is  subject  to  variation,  not  only  with  the  character  of  the 
surface,  but  also  with  the  hydraulic  radius  and  slope. 

Prob.  1  1  8.  Compute  the  value  of  c  for  a  circular  masonry 
conduit  4  feet  in  diameter  which  delivers  29  cubic  feet  per 
second  when  running  half  full,  its  slope  or  grade  being  1.5  feet 
in  1000  feet.  Ans.  119. 


218 


FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 


ARTICLE  95.  CIRCULAR  CONDUITS,  FULL  OR  HALF  FULL. 

When  a  circular  conduit  of  diameter  d  runs  either  full  or 
half  full  of  water  the  hydraulic  radius  is  %d,  and  the  formula 
for  mean  velocity  is 

v  =  ctfrs  =  c.$  Vds. 

The  velocity  can  then  be  computed  when  c  is  known,  and  for 
this  purpose  the  following  table  gives  SMITH'S  values  of  c  for 

TABLE  XVII.    COEFFICIENTS   FOR  CIRCULAR   CONDUITS. 


Diameter 
in 
Feet. 

Velocity  in  Feet  per  Second. 

X 

2 

3 

4 

6 

IO 

15 

I. 

96 

IO4 

109 

112 

116 

121 

124 

1-5 

103 

III 

116 

II9 

123 

129 

132 

2. 

109 

116 

121 

124 

129 

134 

138 

2.5 

H3 

120 

125 

128 

133 

139 

143 

3- 

117 

124 

128 

132 

136 

143 

147 

3-5 

120 

127 

131 

135 

139 

I46 

151 

4- 

123 

130 

134 

137 

142 

150 

155 

5- 

128 

134 

139 

142 

147 

155 

6. 

132 

138 

142 

145 

150 

7- 

135 

141 

145 

149 

153 

8. 

137 

143 

I48 

151 

pipes  and  conduits  having  quite  smooth  interior  surfaces,  and 
no  sharp  bends.*     The  discharge  per  second  then  is 

q  —  av  =  c  .  ^a  Vds, 

in  which  a  is  either  the  area  of  the  circular  cross-section  or  one 
half  that  section,  as  the  case  may  be. 

To  use  this  table  a  tentative  method  must  be  employed, 


*  Hydraulics,  p.  271. 


ART.  95.]  CIRCULAR   CONDUITS,  FULL   OR  HALF  FULL.  219 

since  c  depends  upon  the  velocity  of  flow.  For  this  purpose 
there  may  be  taken  roughly, 

mean  c  =  125, 

and  then  v  may  be  computed  for  the  given  diameter  and  slope  ; 
a  new  value  of  c  is  then  taken  from  the  table  and  a  new  v  com- 
puted ;  and  thus,  after  two  or  three  trials,  the  probable  mean 
velocity  of  flow  is  obtained.  The  value  of  d  must  be  expressed 
in  feet. 

For  example,  let  it  be  required  to  find  the  velocity  and  dis- 
charge of  a  semicircular  conduit  of  6  feet  diameter  when  laid 
on  a  grade  of  o.i  feet  in  100  feet.  First, 


v  —  125  X  £  V6  X  o.oo i  ==  4.8  feet. 
For  this  velocity  the  table  gives  147  for  c ;  hence 

v  —  147  X  i  1/0.006  =  5.7  feet. 
Again,  from  the  table  c  =  1 50,  and 


v  —  150  X  i  Vo.oo6  =  5.8  feet. 

This  shows  that  150  is  a  little  too  large;  for  c  =  149.5,  v  is 
found  to  be  5.79  feet  per  second,  which  is  the  final  result. 
The  discharge  per  second  now  is 

q  =  0.7854  X  £  X  36  X  5.79  =  81.9  cubic  feet, 
which  is  the  probable  flow  under  the  given  conditions. 

To  find  the  diameter  of  a  circular  conduit  to  discharge  a 
given  quantity  under  a  given  slope,  the  area  a  is  to  be  ex- 
pressed in  terms  of  d  in  the  above  equation,  which  is  then  to 
be  solved  for  d',  thus,  for  a  conduit  which  runs  full, 


d 

and  for  one  which  is  half  full 

d 


( **  y 
Wvy ' 

oil 

f  ifr  y 
"  W  «Gl  • 


220  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIIL 

Here  c  at  first  may  be  taken  as  125  ;  then  d  is  computed,  and 
the  approximate  velocity  of  flow  is 


-0.7854^' 

by  which  a  value  of  c  is  selected  from  the  table,  and  the  com- 
putation is  then  repeated  until  the  corresponding  values  of  c 
and  v  are  found  to  closely  agree. 

As  an  example  of  the  determination  of  diameter  let  it  be  re- 
quired to  find  d  when  q  =  81.9  cubic  feet  per  second,  s  =  o.ooi, 
and  the  conduit  runs  full.  For  ^=125  the  formula  gives  d  —  4.9 
feet,  whence  v  =  4.37  feet  per  second.  From  the  table  c  may 
be  now  taken  as  142,  and  repeating  the  computation  d  —  4.64 
feet,  whence  v  —  4.84  feet  per  second,  which  requires  no  further 
change  in  the  value  of  c.  As  the  tabular  coefficients  are  based 
upon  quite  smooth  interior  surfaces,  such  as  occur  only  in  new, 
clean  iron  pipes,  or  with  fine  cement  finish,  it  might  be  well  to 
build  the  conduit  5  feet  or  60  inches  in  diameter.  It  is  seen 
from  the  previous  example  that  a  semicircular  conduit  of  6 
feet  diameter  carries  the  same  amount  of  water  as  is  here  pro- 
vided for. 

A  circular  conduit  running  full  of  water  is  a  long  pipe,  and 
all  the  formulas  and  methods  of  Arts.  80  and  81  can  be  applied 
also  to  their  discussion.  By  comparing  the  formulas  of  velocity 
for  pipes  and  conduits, 


it  is  seen  that 


-  , 

in  which  f  is  to  be  taken  from  Table  XVI.  Values  of  c  com- 
puted in  this  manner  will  not  generally  agree  closely  with  the 
coefficients  of  SMITH,  partly  because  the  values  of  /  are  given 


ART.  96.]  CIRCULAR   CONDUITS,  PARTLY  FULL.  221 

only  to  three  decimal  places,  and  partly  because  Table  XVI 
was  constructed  by  regarding  other  discussions.  An  agree- 
ment within  5  per  cent  in  mean  velocities  deduced  by  different 
methods  is  all  that  can  generally  be  expected  in  conduit  com- 
putations, and  if  the  actual  discharge  agrees  as  closely  as  this 
with  the  computed  discharge,  the  designer  can  be  considered 
as  a  fortunate  man. 

All  of  the  laws  deduced  in  the  last  chapter  regarding  the 
relation  between  diameter  and  discharge,  relative  discharging 
capacity,  etc.,  hence  apply  equally  well  to  circular  conduits 
which  run  either  full  or  half  full.  Ancl  if  the  conduit  be  full 
it  matters  not  whether  it  be  laid  truly  to  grade  or  whether  a 
portion  of  it  be  under  pressure,  since  in  either  case  the  slope 
s  is  the  total  fall  h  divided  by  the  total  length.  Usually,  how- 
ever, the  word  conduit  implies  a  uniform  slope  for  consider- 
able distances,  and  in  this  case  the  hydraulic  gradient  coincides 
with  the  surface  of  the  flowing  water. 

Prob.  119.  Find  the  discharge  of  a  conduit  when  running 
full,  its  diameter  being  6  feet  and  its  fall  9.54  feet  in  one  mile. 

Prob.  1 2.0.  Find  the  diameter  of  a  conduit  to  deliver  when 
running  full  16  500000  gallons  per  day,  its  slope  being  0.00016. 

ARTICLE  96.  CIRCULAR  CONDUITS,  PARTLY  FULL. 

Let  a  circular  conduit  with  the  slope  s  be  partly  full  of 
water,  its  cross-section  being  a  and  hydraulic  radius  r.  Then 
the  mean  velocity  of  flow  is 

v  =  c  Vrs, 
and  the  discharge  per  second  is 

q  =  av  =  c.  a  Vrs. 

The  mean  velocity  is  hence  proportional  to  Vr  and  the  dis- 
charge to  a  Yr,  provided  that  c  be  a  constant.  Since,  how- 
ever, c  varies  slightly  with  r,  this  law  of  proportionality  is 
approximate. 


222 


FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIIL 


When  a  circular  conduit  of  diameter  d  runs  either  full  or 
half  full  its  hydraulic  radius  is  \d  (Art.  93).     If  it  is  filled  to 
the  depth  d' ',  the  wetted  perimeter  is 

2d' -d 


p  —  ^nd-\-  d  arc  sin 


d 


and  the  sectional  area  of  the  water  surface  is 


FIG.  67.  a  ~ 

From  these  /  and  a  can  be  computed,  and  then  r  is  found  by 
dividing  a  by/.  The  following  table  gives  values  of  /,  #,  and 
r  for  a  circle  whose  diameter  is  unity  for  different  depths  of 
water.  To  find  from  it  the  hydraulic  radius  for  any  other  cir- 

TABLE  XVIII.     CROSS-SECTIONS  OF  CIRCULAR  CONDUITS. 


Depth 
df 

Wetted 
Perimeter 

P 

Sectional  Area 
a 

Hydraulic 
Radius 

r 

Velocity 

Vr 

Discharge 

«*£ 

Full            I.O 

3.142 

0.7854 

0.25 

0.5 

0-393 

.  0.95 

2.6QI 

0.7708 

0.286 

0-535 

.413 

0.9 

2.498 

0-7445 

0.298 

0.546 

.406 

0.81 

2.240 

0.6815 

0.3043 

0.552 

•  376 

0.8 

2.214 

0-6735 

0.3042 

0.552 

•  372 

0.7 

1.983 

0.5874 

0.296 

0-544 

.320 

0.6 

1.772 

0.4920 

0.278 

0.527 

•259 

Half  Full  o'.  5 

I-57I 

0.3927 

0.25 

0.5 

.196 

0.4 

1.369 

0.2934 

O.2I4 

0.463 

.136 

0.3 

I-I59 

0.1981 

O.I7I 

0.414 

.0820 

0.2 

0.927 

0.1118 

O.I2I 

0.348 

.0389 

O.I 

0.643 

0.0408 

0.0635 

O.252 

.0103 

Empty      o.o 

0.0 

o.o 

0.0 

0.0 

o.o 

cle  it  is  only  necessary  to  multiply  the  tabular  values  of  r  by 
the  given  diameter  d.  The  table  shows  that  the  greatest  value 
of  the  hydraulic  radius  occurs  when  d'  —  o.$id,  and  that  it  is 
but  little  less  when  d'  =  o.SV. 


ART.  96.]  CIRCULAR   CONDUITS,  PARTLY  FULL.  22$ 

In  the  fifth  and  sixth  columns  of  the  table  are  given  values 
of  Vr  and  a  Vr  for  different  depths  in  the  circle  whose  diame- 
ter is  unity;  these  are  approximately  proportional  to  the 
velocity  and  discharge  which  occur  at  those  depths  in  a  circle 
of  any  size.  The  table  shows  that  the  greatest  velocity  occurs 
when  the  depth  of  the  water  is  about  eight-tenths  of  the  di- 
ameter, and  that  the  greatest  discharge  occurs  when  the  depth 
is  about  0.95^,  or  £$-ths  of  the  diameter. 

By  the  help  of  the  above  table  the  velocity  and  discharge 
may  be  computed  when  c  is  known,  but  it  is  not  possible  on 
account  of  the  lack  of  experimental  knowledge  to  state  precise 
values  of  c  for  different  values  of  r  in  circles  of  different  sizes. 
However,  it  is  known  that  an  increase  in  r  increases  <:,  and  that 
a  decrease  in  r  decreases  c.  The  following  experiments  of 
DARCY  and  BAZIN  show  the  extent  of  this  variation  for  a  semi- 
circular conduit  of  4.1  feet  diameter,  and  they  also  teach  that 
the  nature  of  the  interior  surface  greatly  influences  the  values 
of  c.  Two  conduits  were  built  each  with  a  slope  s  =  0.0015 
and  d  =  4.1  feet.  One  was  lined  with  neat  cement,  and  the 
other  with  a  mortar  made  of  cement  with  one-third  fine  sand. 
The  flow  was  allowed  to  occur  with  different  depths,  and  the 
discharges  per  second  were  gauged  by  means  of  orifices ;  this 
enabled  the  velocities  to  be  computed,  and  from  these  the 
values  of  c  were  found.  The  following  are  a  portion  of  the  re- 
sults obtained,  d'  denoting  the  depth  of  water  in  the  conduit, 
and  all  dimensions  being  in  feet :  * 

For  cement  lining  For  mortar  lining 


d' 

r 

V 

c 

d' 

r 

V 

c 

2.05 

1.029 

6.06 

154 

2.04 

i.  022 

5-55 

142 

1.83 

0.949 

5-75 

152 

i.  80 

0.941 

5.20 

138 

1  .61 

0.867 

5-29 

147 

1.69 

0.900 

4.94 

135 

1-34 

0.750 

4.87 

145 

1.41 

0.787 

4-51 

131 

1.03 

0.605 

4.16 

138 

1.09 

0.635 

3-87 

125 

0.83 

0.503 

3-72 

136 

0.88 

0.529 

3-43 

122 

0-59 

0.366 

3.02 

129 

0.61 

0-379 

2.87 

120 

*  SMITH'S  Hydraulics,  p.  176. 


224  FLO  W  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

It  is  here  seen  that  c  decreases  quite  uniformly  with  r,  and 
that  the  velocities  for  the  mortar  lining  are  8  or  10  per  cent 
less  than  for  the  neat  cement  lining. 

The  value  of  the  coefficient  c  for  these  experiments  may  be 
roughly  expressed  by  the  formula 


-  d'), 

in.  which  c'  is  the  coefficient  for  the  conduit  when  running  half 
full.  How  this  will  apply  to  different  diameters  and  velocities 
is  not  known  ;  when  d'  is  greater  than  o.%d  it  will  probably 
prove  incorrect.  In  practice,  however,  computations  on  the 
flow  in  partly  filled  conduits  are  of  rare  occurrence. 

Prob.  121.  Compute  the  hydraulic  radius  for  a  circular 
conduit  when  it  is  three-fourths  filled  with  water,  and  also  the 
mean  velocity  if  it  be  lined'with  pure  cement  and  laid  on  a 
grade  of  0.15  per  100,  the  diameter  being  4.1  feet. 

ARTICLE  97.  OPEN  RECTANGULAR  CONDUITS. 

In  designing  an  open  rectangular  trough  or  conduit  to 
carry  water  there  is  a  certain  ratio  of  breadth  to  depth  which 
is  most  advantageous,  because  that  thereby  either  the  dis- 
charge is  the  greatest  or  the  least  amount  of  material  is  re- 
quired for  its  construction.  This  advantageous  proportion  is 
the  one  which  offers  the  least  frictional  resistance  to  the  flow  ; 
in  a  very  wide  and  shallow  trough  the  friction  would  be  great, 
and  the  same  would  be  the  case  in  one  of  small  width  and 
large  depth.  It  is  now  to  be  shown  that  the  least  friction, 
and  hence  the  best  proportions,  results  when  the  width  is 
double  of  the  depth. 

The  head  lost  in  friction  is  directly  proportional  to  the 
wetted  perimeter  and  inversely  proportional  to  the  area  of  the 
water  cross-section  (Art.  94).  In  order  that  this  "may  be  "the 


ART.  97.]  OPEN  RECTANGULAR   CONDUITS.  22$ 

least  possible,  the  wetted  perimeter  should  be  a  minimum  for 
a  given  area,  or  the  area  should  be  a  maximum  for  a  given 
wetted  perimeter.  But  the  ratio  of  the  area  to  the  perimeter 
is  the  hydraulic  radius 

which  therefore  is  to  be  a  maximum,  subject  to  the  other  con- 
ditions of  the  problem,  in  order  to  secure  the  most  advantage- 
ous cross-section.  This  is  an  approximate  general  rule,  appli- 
cable to  all  kinds  of  channels,  and  it  is  plain  that  the  circle 
fulfils  the  requirement  in  a  higher  degree  than  any  other 
figure. 

For  an  open  rectangular  conduit  of  breadth  b  and  depth  d 
the  value  of  the  hydraulic  radius  is 

bd 

Y  '==-  ~i — ; T  • 


If  it  be  required  to  find  the  most  advantageous  section  for  a 
given  wetted  perimeter,  this  may  be  written 


2P 


and  this  is  seen  to  be  a  maximum  when  b  =  \p,  that  is,  when 
b  =  2d,  or  the  breadth  is  double  the  depth.  If,  however,  it  be 
required  to  determine  the  most  advantageous  section  for  a 
given  area,  the  value  of  the  hydraulic  radius  may  be  written 


and  by  equating  the  first  derivative  to  zero,  there  is  found 
#a  =  2#,  from  which  ff  =  2bdt  or  b  =  2dy  as  before. 


226  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIIL 

Again,  if  it  be  required  to  find  the  most  advantageous  sec- 
tion to  carry  q  cubic  feet  of  water  per  second,  the  hydraulic 
radius 

_      bd 
~  b  +  2d  - 

is  to  be  made  a  maximum,  subject  to  the  condition 


/—  /    b*d* 

q  =  c .  a  Vrs  =P  cs*  \  /  T— ; -j  • 

y   b-}-  2d 


Regarding  c  as  a  constant,  the  values  of  b  and  d  which  render 
r  a  maximum  can  be  ascertained  by  the  rules  of  the  higher 
analysis,  and  there  is  also  found  for  this  case  the  relation  b  =  2dy 
or  the  breadth  is  double  the  depth. 

The  velocity  and  discharge  through  a  rectangular  conduit 
are  expressed  by  the  general  equations 

v  =  c  Vrs,         q  =  av, 

and  are  computed  without  difficulty  for  any  given  case  when 
the  coefficient  c  is  known.  To  ascertain  this,  however,  is  not 
easy  on  account  of  the  lack  of  experiments  by  which  alone  its 
value  can  be  ascertained.  When  the  depth  of  the  water  in  the 
conduit  is  one-half  of  its  width,  thus  giving  the  most  advan- 
tageous section,  the  values  of  c  for  smooth  interior  surfaces 
may  be  estimated  from  the  table  in  Art.  96  for  circular  con- 
duits, although  c  is  probably  smaller  for  rectangles  than  for 
circles  of  equal  area.  When  the  depth  of  the  water  is  less  or 
greater  than  \d,  it  must  be  remembered  that  c  increases  with  r. 
The  value  of  c  also  is  subject  to  slight  variations  with  the  slope 
s,  and  to  great  variations  with  the  degree  of  roughness  of  the 
surface. 

The  following  table,  derived  from  SMITH'S  discussion  of  the 
experiments  of  DARCY  and  BAZIN,  gives  values  of  c  for  a  num- 


TRAPEZOIDAL    SECTIONS. 


227 


ART.  98.] 

ber  of  wooden  and  masonry  conduits  with  rectangular  sections, 

all  of  which  were  laid  on  the  grade  of  0.49  feet  per  100,  or 

TABLE  XIX.  COEFFICIENTS  FOR  RECTANGULAR  CONDUITS. 


Unplaned  Plank. 
b  =  3-93  feet- 

Unplaned  Plank. 
b  =  6.53  feet. 

Pure  Cement. 
b  =  5.94  feet. 

Brick. 
b  =  6.27  feet. 

d 

c 

d 

c 

d 

e 

d 

c 

0.27 

99 

0.20 

89 

0.18 

116 

0.20 

89 

.41 

108 

•30 

101 

.28 

125 

•31 

98 

•67 

112 

.46 

109 

•43 

132 

.49 

104 

.89 

114 

.60 

H3 

.56 

135 

•57 

105 

1.  00 

114 

.72 

116 

.63 

136 

•65 

104 

I.I9 

116 

.78 

116 

.69 

136 

•71 

106 

1.29 

117 

.89 

118 

.80 

137 

.85 

107 

I.46 

118 

•94 

120 

.91 

138 

•97 

no 

s  =  0.0049.  The  great  influence  of  roughness  of  surface  in 
diminishing  the  coefficient  is  here  plainly  seen.  For  masonry 
conduits  with  hammer-dressed  surfaces  c  may  be  as  low  as  60 
or  50,  particularly  when  covered  with  moss  and  slime. 

Prob.  122.  Compare  the  discharge  of  a  trough  I  X  3  feet 
with  that  of  two  troughs  each  I  X  2  feet. 

Prob.  123.  Find  the  size  of  a  trough,  whose  width  is  double 
its  depth,  which  will  deliver  125  cubic  feet  per  minute  when  its 
slope  is  0.002,  taking  c  as  100.  Ans.  d  —  0.64  feet. 

ARTICLE  98.  TRAPEZOIDAL  SECTIONS. 

Ditches  and  conduits  are  often  built  with  a  bottom  nearly 
flat  and  with  side  slopes,  thus  forming  a  trapezoidal  section. 
The  side  slope  is  fixed  by  the  nature  of  the  soil  or  by  other 
circumstances,  the  grade  is  given,  and  it  may  be  then  required 
to  ascertain  the  relation  between  the  bottom  width  and  the 
depth  of  water,  in  order  that  the  section  shall  be  the  most  ad- 
vantageous. This  can  be  done  by  the  same  reasoning  as  used 


228  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

for  the  rectangle  in  the  last  article,  but  it  may  be  well  to  em- 
ploy a  different  method,  and  thus  be  able  to  consider  the  sub- 
ject in  a  new  light. 

Let  the  trapezoidal  channel  have  the  bottom  width  b,  the 
depth  d,  and  let  0  be  the  angle  made  by  the  side  slopes  with  the 

horizontal.  Let  it  be  required  to 
discharge  q  cubic  feet  per  second  ; 
then 


^Nili»p 

FIG.  68.  q  =  caVrs. 

"Now  the  most  advantageous  proportions  may  be  said  to  be 
those  that  \vill  render  the  cross-section  a  a  minimum,  for  thus 
the  least  excavation  will  be  required.  The  above  equation  may 
be  written 

<f        a3 

7s=~p' 

In  this/  is  to  be  replaced  by  its  value  in  terms  of  a  and  d,  and 
then  the  value  of  d  is  to  be  found  which  renders  a  a  minimum. 
For  this  purpose  the  figure  gives 

d  cot  0) ; 


- 

sin  6      a*      Vsin 
from  which  the  equation  becomes 


Obtaining  the  first  derivative  of  a  with  respect  to  d,  and  equat- 
ing it  to  zero,  there  is  found 


6  -  cot  0\d*  =  a  ; 
in  6  / 


8 
and  replacing  for  a  its  value,  there  results 


(70 


ART.  98.]  TRAPEZOIDAL   SECTIONS.  2 29 

which  is  the  relation  that  gives  the  most  advantageous  cross- 
section.  If  6  =:  90°,  the  trapezoid  becomes  a  rectangle,  and 
b  =  2d,  as  previously  deduced.  As  c  has  been  regarded  as  a 
constant  in  this  investigation,  the  conclusion  is  not  a  rigorous 
one,  although  it  may  be  safely  followed  in  practice.  It  is  to  be 
expected,  as  in  all  cases  of  a  maximum,  that  quite  considerable 
variations  in  the  ratio  b  :  d  may  occur  without  materially  affect- 
ing the  value  of  a. 

When  the  value  of  c  is  known,  the  general  formulas^  =  c-Vrs 
and  q  =  av  may  be  used  to  obtain  a  rough  approximation  to 
the  discharge.  The  formula  of  KUTTER  (Art.  101)  may  be  used 
to  determine  c  when  the  nature  of  the  bed  of  the  channel  is 
known.  In  any  important  case,  however,  computations  cannot 
be  trusted  to  give  reliable  values  of  the  discharge  on  account 
of  the  uncertainty  regarding  the  coefficient,  and  an  actual 
gauging  of  the  flow  should  be  made.  This  is  best  effected  by 
a  weir,  but  if  that  should  prove  too  expensive,  the  methods 
explained  in  Chap.  IX  may  be  employed  to  give  more  precise 
results  than  can  usually  be  determined  by  any  computation. 

The  problem  of  determining  the  size  of  a  trapezoidal 
channel  to  carry  a  given  quantity  of  water,  does  not  require  c 
to  be  determined  so  closely.  For  this  purpose  the  following 
values  may  be  used,  the  lower  ones  being  for  small  cross-sec- 
tions with  rough  and  foul  surfaces,  and  the  higher  ones  for 
quite  smooth  surfaces : 

For  unplaned  plank,  c  =  100  to  120 

For  smooth  masonry,  c  =    90  to  1 10 

For  clean  earth,  c  =    60  to    80 

For  stony  earth,  c  =    40  to    60 

For  rough  stone,  c  =    35  to    50 

For  earth  foul  with  weeds,  c  =    30  to    50 


230  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

To  solve  this  problem,  let  a  and  /  be  replaced  by  their  values 
in  terms  of  b  and  d.     The  discharge  then  is 


.  _  x_  ,,   — ot  0)s  sin  9 

'V ^  sin  0  +  2aT ' 

Now  when  q,  c,  0,  and  s  are  known,  the  equation  contains  two 
unknown  quantities,  b  and  d.  If  the  section  is  to  be  the  most 
advantageous,  b  can  be  replaced  by  its  value  in  terms  of  d  as 
above  found,  and  the  equation  then  has  but  one  unknown.  Or 
in  general,  if  b  =  md,  where  m  is  any  assumed  number,  the 
solution  gives 

,5  _      q*(m  sin  9  -\-  2) 

"" ~       '  '  ^\  3         •  /i  • 

r  sin  ^ 


For  the  particular  case  where  the  side  slopes  are  I  to  I  or 
6  —  45°,  and  the  bottom  width  is  to  be  equal  to  the  water 
depth,  or  m  =  i,  this  becomes 


=  0.863 


These  formulas  are  analogous  to  those  for  finding  the  diameter 
of  pipes  and  circular  conduits,  and  the  numerical  operations 
are  in  all  respects  similar.  It  is  plain  that  by  assigning  dif- 
ferent values  to  m  numerous  sections  may  be  determined 
which  will  satisfy  the  imposed  conditions,  and  usually  the  one 
is  to  be  selected  that  will  give  both  a  safe  velocity  and  a 
minimum  cost.  In  Art.  103  will  be  found  an  example  of  the 
determination  of  the  size  of  a  trapezoidal  canal. 

Prob.  124.  If  the  value  of  c  is  71,  compute  the  depth  of  a 
trapezoidal  section  to  carry  200  cubic  feet  of  water  per  second, 
6  being  45°,  the  slope  s  being  o.ooi,  and  the  bottom  width 
being  equal  to  the  depth.  Compute  also  the  mean  velocity 
for  the  section. 


ART.  99-]  HORSE-SHOE   CONDUITS.  231 

ARTICLE  99.  HORSE-SHOE  CONDUITS. 

In  Fig.  69  is  given  an  outline  cross-section  of  the  Sudbury 
conduit,  the  flow  of  which  was  gauged  by  FTELEY  and 
STEARNS,  whose  discussions  have  determined  a 
formula  for  its  mean  velocity.  The  section 
consists  of  a  part  of  a  circle  of  9.0  feet  diameter, 
having  an  invert  of  13.22  feet  radius,  whose 
span  is  8.3  feet  and  depression  0.7  feet,  the 
axial  depth  of  the  conduit  being  7.7  feet.  The 
conduit  is  lined  with  brick,  having  cement  joints  one  quarter  of 
an  inch  thick.  The  flow  was  allowed  to  occur  with  different 
depths,  for  each  of  which  the  discharge  was  determined  by 
weir  measurement.  A  discussion  of  the  results  led  to  the 
conclusion  that  in  the  portion  with  the  brick  lining  the  coeffi- 
cient c  had  the  value  I27r°12  when  r  is  in  feet,  and  hence 

v  =  12??"-12  V~rs  =  I27r°-62s°-*  .....     (72) 


In  a  portion  of  the  conduit  where  the  brick  lining  was  coated 
with  pure  cement  the  coefficient  was  found  to  be  from  7  to  8 
per  cent  greater  than  127.  In  another  portion  where  the  brick 
lining  was  covered  with  a  cement  wash  laid  on  with  a  brush 
the  coefficient  was  from  I  to  3  per  cent  greater.  For  a  long 
tunnel  in  which  the  rock  sides  were  ragged,  but  with  a  smooth 
cement  floor,  it  was  found  to  be  about  40  per  cent  less.* 

These  results  clearly  show  that  the  coefficient  c  increases 
with  r,  and  that  it  is  greatly  influenced  by  the  nature  of  the 
interior  surfaces.  For  sections  of  smaller  area  than  that  above 
given  the  value  of  c  is  undoubtedly  less  than  1277-°",  and  for 
those  of  larger  area  it  is  greater  ;  the  extent  of  variation  may 
perhaps  be  inferred  from  the  table  in  Art.  95.  The  general 

*  Transactions  American  Society  Civil  Engineers  1883,  vol.  xii.  p.  114. 


232  FLOW  IN  CONDUITS  AND   CANALS.      [CHAP.  VIII, 

slope  of  the  Sudbury  conduit  is  about  one  foot  per  mile,  and  c 
is  also  subject  to  variation  with  s,  as  well  as  with  the  tempera- 
ture of  the  water.  Although  the  above  formula  is  a  special 
one,  applicable  to  a  single  conduit,  it  is  nevertheless  of  great 
value,  as  it  presents  the  only  existing  evidence  regarding  the 
coefficients  for  large  aqueducts. 

Prob.  125.  The  actual  discharge  of  the  Sudbury  conduit  is 
about  60  080  ooo  gallons  per  24  hours  when  the  water  is  4  feet 
deep,  a  being  33.31  sq.  feet,  p  =  15.21  feet,  and  s  =  0.0001895. 
Compute  the  discharge  by  the  use  of  the  above  formula. 

ARTICLE  100.  LAMPE'S  FORMULA. 

The  formula  given  in  Art.  91  for  the  mean  velocity  of  flow 
in  long  circular  pipes  can  be  also  applied  to  conduits  with  very 
smooth  surfaces.  Replacing  for  the  ratio  h  -r-  /  the  slope  sr 
and  for  d  its  equivalent  4?-,  it  becomes 


v  =  203^°-69V-555.  .     .     .     .    .    .    .     (73) 

This  formula  may  be  also  written 

v  =  203r°-I%0-055  Vrsl     .....   (73)' 


in  which  the  quantity  preceding  the  radical  in  the  second  mem- 
ber is  the  coefficient  c.  According  to  this  empirical  expression 
c  increases  both  with  r  and  s,  but  only  slightly  with  the  latter. 
It  is  probable  that  this  formula  represents  quite  accurately  the 
laws  of  flow  in  conduits,  but  the  varying  degree  of  roughness 
of  surface  is  not  taken  into  account  by  it,  so  that  in  general  it 
can  only  be  used  to  furnish  approximate  results,  except  for  the 
case  of  metal  pipes  or  similar  smooth  surfaces.  For  this  pur- 
pose the  formulas  for  q  and  d,  given  in  Art.  91,  may  be  directly 
used  for  circular  sections.  It  is  probable  that  future  researches. 
may  show  that  a  formula  similar  to  the  above  may  fairly  repre- 


ART.  ioi.]  KUTTER'S  FORMULA.  233 

sent  all  cases,  the  constant  203  being  varied  with  the  roughness 
of  the  surface. 

Prob.  126.  Solve  Prob.  125  by  the  use  of  LAMPE's  formula, 
and  compare  the  error  of  the  result  with  that  as  deduced  by 
the  special  formula  for  the  conduit. 


ARTICLE  ioi.  KUTTER'S  FORMULA. 

The  researches  of  GANGUILLET  and  KUTTER  have  furnished 
a  general  expression  for  the  coefficient  c  in  the  formula  for 
mean  velocity, 

v  =  c  Vlrsl 

by  which  its  value  can  be  computed  for  any  given  case  when 
the  nature  of  the  interior  surface  is  known.  This  expression  is, 
for  English  measures^ 

1.811  >- 

-—  +  41.65 

(74) 


in  which  a  is  an  abstract  number  whose  value  depends  only 
upon  the  roughness  of  the  surface,  and 

a  =  0.009  for  well-planed  timber ; 

a  =  o.oio  for  neat  cement ; 

a  =  o.oii  for  cement  with  one-third  sand; 

a  =  0.012  for  unplaned  timber ; 

a  =  0.013  for  ashlar  and  brickwork; 

a  =  0.015  for  unclean  surfaces  in  sewers  and  conduits; 

a  =  0.017  for  rabble  masonry; 

a  =  0.020  for  canals  in  very  firm  gravel ; 


234  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

a  —  0.025  for  canals  and  rivers  free  from  stones  and  weeds; 
a  =  0.030  for  canals  and  rivers  with  some  stones  and  weeds ; 
a  —  0.035  for  canals  and  rivers  in  bad  order. 

By  inserting  this  value  of  c  in  the  formula  for  v,  the  mean 
velocity  is  made  to  depend  upon  r,  s,  and  the  roughness  of  the 
surface. 

The  formula  of  KUTTER  has  received  a  wide  acceptance  on 
account  of  its  application  to  all  kinds  of  surfaces.  Notwith- 
standing that  it  is  purely  empirical,  and  hence  not  perfect,  it  is 
to  be  regarded  as  a  formula  of  great  value,  so  that  no  design 
for  a  conduit  or  channel  should  be  completed  without  employ- 
ing it  in  the  investigation,  even  if  the  final  construction  be  not 
based  upon  it.  In  sewer  work  it  is  extensively  employed,  a 
being  taken  as  about  0.015.  The  formula  shows  that  c  always 
increases  with  r,  that  it  decreases  with  s  wheni  r  is  greater  than 
3.28  feet,  and  that  it  increases  with  s  when  r  is  less  than  3.28 

T  8 1 1 
feet.     When  r  equals  3.28  feet  the  value  of  c  is  simply  — '— . 

It  is  not  likely  that  future  investigations  will  confirm  these  laws 
of  variation  in  all  respects. 

In  the  following  articles  are  given  values  of  c  for  a  few 
cases,  and  these  might  be  greatly  extended,  as  has  been  done 
by  KUTTER  and  others.  But  this  is  scarcely  necessary  except 
for  special  lines  of  investigation,  since  for  single  cases  there 
is  no  difficulty  in  directly  computing  it  for  given  data.  For 
instance,  take  a  rectangular  trough  of  unplaned  plank  3.93 
feet  wide  on  a  slope  of  0.0049,  the  water  being  1.29  feet  deep. 
Here 

s  =  0.0049 
and 

'=ff+4i  =  a779feet 


ART.  102.]  SEWERS.  235 

Then  a  being  0.012,  the  value  of  c  is  found  to  be 

1.8 1 1  >.      ,    0.00281 
-4-  41.65  +  — 

0.012  0.0049 

0.012     I        IT       .     0.0028A 


.65  +  - 

0.0049 ' 

The  data  here  used  are  taken  from  the  table  in  Art.  97,  where 
the  actual  value  of  c  is  given  as  117;  hence  in  this  case 
KUTTER's  formula  is  about  5  per  cent  in  excess.  As  a  second 
example,  the  following  data  from  the  same  table  will  be  taken : 
a  rectangular  conduit  in  pure  cement,  £—  5.94  feet,  d  =  0.91 
feet,  s  —  0.0049.  Here  a  =  o.oio,  and  r  =  0.697  feet.  Insert- 
ing all  values  in  the  formula,  there  is  found  c  =  148,  which  is 
8  per  cent  greater  than  the  true  value,  138.  Thus  is  shown  the 
fact  that  errors  of  5  and  10  per  cent  are  to  be  regarded  as  com- 
mon in  calculations  on  the  flow  of  water  in  conduits  and  canals. 

Prob.  127.  Compute  by  KUTTER'S  formula  the  discharge 
for  the  data  in  Prob.  125. 

ARTICLE  102.  SEWERS. 

Sewers  smaller  in  diameter  than  18  inches  are  always  cir- 
cular in  section.  When  larger  than  this  they  are  built  with 
the  section  either  circular,  egg-shaped,  or  of  the  horseshoe 
form.  The  last  shape  is  very  disadvantageous  when  a  small 
quantity  of  sewage  is  flowing,  for  the  wetted  perimeter  is  then 
large  compared  with  the  area,  the  hydraulic  radius  is  small,  and 
the  velocity  becomes  low,  so  that  a  deposit  of  the  foul  materials 
results.  As  the  slope  of  sewer  lines  is  often  very  slight,  it  is 
important  that  such  a  form  of  cross-section  should  be  adopted 
to  render  the  velocity  of  flow  sufficient  to  prevent  this  deposit. 
A  velocity  of  2  feet  per  second  is  found  to  be  about  the  mini- 
mum allowable  limit,  and  4  feet  per  second  need  not  be  usu- 
ally exceeded. 


FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIIL 


The  egg-shaped  section  is  designed  so  that  the  hydraulic 
radius  may  not  become  small  even  when  a  small  amount  of 

sewage    is   flowing.     One    of    the 
most  common  forms  is  that  shown 
C    in  Fig.  70,  where  the  greatest  width 
DD    is  two-thirds   of    the    depth 
HM.     The  arch  DHD  is  a  semi- 
circle described  from  A  as  a  centre. 
The  invert  LML  is  a  portion  of 
a  circle  described  from  B  as  a  cen- 
FIG.  70.  tre,  the  distance  BA  being  three- 

fourths  of  DD  and  the  radius  BM  being  one-half  of  AD. 
Each  side  DL  is  described  from  a  centre  C  so  as  to  be  tangent 
to  the  arch  and  invert.  These  relations  may  be  expressed 
more  concisely  by 

HM  =  ii/7,      AB  =  f/7,       BM  =  £/7,       CL  =  i£/7, 
in  which  D  is  the  horizontal  diameter  DD. 

Computations  on  egg-shaped  sewers  are  usually  confined  to 
three  cases,  namely,  when  flowing  full,  two-thirds  full,  and  one- 
third  full.  The  values  of  the  sectional  areas,  wetted  perimeters> 
and  hydraulic  radii  for  these  cases,  as  given  by  FLYNN,*  are 

a  p                    r 

Full                          1. 1485/7*  3.965/7  0.2897/7 

Two-thirds  full       0.7558/7*  2.394/7  0.3157/7 

One-third  full         0.2840/7*  1-375/7  0.2066/7 

This  shows  that  the  hydraulic  radius,  and  hence  the  velocity, 
is  but  little  less  when  flowing  one-third  full  than  when  flowing 
with  full  section. 

Egg-shaped  sewers  and  small  circular  ones  are  formed  by 
laying  consecutive  lengths  of  clay  or  cement  pipe  whose  interior 


Van  Nostrand's  Magazine,  1883,  vol.  xxviii.  p.  138. 


ART.  102.]  SEWERS.  -237 

surfaces  are  quite  smooth  when  new,  but  may  become  foul  after 
use.  Large  sewers  of  circular  section  are  made  of  brick,  and 
are  more  apt  to  become  foul  than  smaller  ones.  In  the  separate 
system,  where  systematic  flushing  is  employed  and  the  pipes 
are  small,  foulness  of  surface  is  not  so  common  as  in  the  com- 
bined system,  where  the  storm  water  is  alone  used  for  this 
purpose.  In  the  latter  case  the  sizes  are  computed  for  the 
volume  of  storm  water  to  be  discharged,  the  amount  of  sewage 
being  very  small  in  comparison. 

The  discharge  of  a  sewer  pipe  enters  it  at  intervals  along 
its  length,  and  hence  the  flow  is  not  uniform.  The  depth  of 
the  flow  increases  along  the  length,  and  at  junctions  the  size  of 
the  pipe  is  enlarged.  The  strict  investigation  of  the  problem 
of  flow  is  accordingly  one  of  great  complexity.  But  consider- 
ing the  fact  that  the  sewer  is  rarely  filled,  and  that  it  should  be 
made  large  enough  to  provide  for  contingencies  and  future 
extensions,  it  appears  that  great  precision  is  unnecessary.  The 
universal  practice,  therefore,  is  to  discuss  a  sewer  for  the  con- 
dition of  maximum  discharge,  regarding  it  as  a  channel  with 
uniform  flow.  The  main  problem  is  that  of  the  determination 
of  size ;  if  the  form  be  circular,  the  diameter  is  found,  as  in 
Art.  95,  by 


If  the  form  be  egg-shaped  and  of  the  proportions  above  ex- 
plained, the  discharge  when  running  full  is 

q  =  ac  Ws  =  I.I485/7V  1/ 
from  which  the  value  of  D  is  found  to  be 

' 


(4). 

Wy 


Thus  when  q  has  been  determined  and  c  is  known  the  required 
sizes  for  given  slopes  can  be  computed.     The  velocity  should 


FLOW  IN  CONDUITS  AND   CANALS.       [^HAP.  VIII. 


also  be  found   in  order  to  ascertain  if  it  be  high  enough  to 
prevent  deposit  (Art.  108). 

Few  or  no  experiments  exist  from  which  to  directly  deter- 
mine  the  coefficient  c  for  the  flow  in  sewers,  but  since  the  sew- 
age is  mostly  water,  it  may  be  approximately  ascertained  from 
the  values  for  similar  surfaces.  KUTTER'S  formula  has  been 
extensively  employed  for  this  purpose,  using  0.015  f°r  the 
coefficient  of  roughness.  The  following  table  gives  values  of  c 
for  three  different  slopes  and  for  two  classes  of  surfaces.  The 
values  for  the  degree  of  roughness  represented  by  a  =  0.017 

TABLE  XX.  COEFFICIENTS  FOR  SEWERS. 


Hydraulic 
Radius  r 
in  Feet. 

j  —  0.00005 

J  =  O.OOOI 

S  =  O.OI 

a  =  0.015 

a  =  0.017 

a  =  0.015 

a  =  0.017 

a  =  0.015 

68 

a  =  0.017 

0.2 

52 

43 

58 

48 

57 

0-3 

60 

5i 

66 

56 

76 

64 

0.4 

65 

56 

73 

61 

83 

70 

0.6 

76 

65 

82 

70 

90 

76 

0.8 

82 

72 

87 

76 

95 

82 

I. 

88 

77 

92 

80 

99 

87 

1-5 

100 

86 

103 

89 

108 

93 

2. 

106 

94 

108 

96 

in 

99 

3- 

116 

103 

1x8 

104 

11$ 

105 

are  applicable  to  sewers  with  quite  rough  surfaces  of  masonrj,  ; 
those  for  a  =  0.015  are  applicable  to  sewers  with  ordinary 
smooth  surfaces,  somewhat  fouled  or  tuberculated  by  deposits, 
and  are  the  ones  to  be  generally  used  in  computations.  By 
the  help  of  this  table  and  the  general  equations  for  mean 
velocity  and  discharge  all  problems  relating  to  flow  in  sewers 
can  be  readily  solved. 

Prob.  128.  The  grade  of  a  sewer  is  one  foot  in  960,  and  its 
discharge  is  to  be  65  cubic  feet  per  second.  What  is  the  diam- 
eter of  the  sewer  if  circular  ?  Ans.  d  =  4.8  feet. 


ART.  103.] 


DITCHES  AND   CANALS. 


239 


ARTICLE  103.  DITCHES  AND  CANALS. 

Ditches  for  irrigating  purposes  are  of  a  trapezoidal  section, 
and  the  slope  is  determined  by  the  fall  between  the  point 
from  which  the  water  is  taken  and  the  place  of  delivery.  If 
the  fall  is  large  it  may  not  be  possible  to  construct  the  ditch  in 
a  straight  line  between  the  two  points,  even  if  the  topography 
of  the  country  should  permit,  on  account  of  the  high  velocity 
which  would  result.  A  velocity  exceeding  2  feet  per  second 
may  often  prove  injurious  in  wearing  the  bed  of  the  channel 
unless  protected  by  riprap  or  other  lining.  For  this  reason  as 
well  as  for  others  the  alignment  of  ditches  and  canals  is  often 
circuitous. 

The  principles  of  the  preceding  articles  are  sufficient  to 
solve  all  usual  problems  of  uniform  flow  in  such  channels  when 
the  values  of  c  are  known.  These  are  perhaps  best  determined 
by  KUTTER'S  formula,  and  for  greater  convenience  a  table  is 

TABLE  XXI.  COEFFICIENTS  FOR  CHANNELS  IN  EARTH. 


Hydraulic 
Radius  r 
in  Feet. 

s  =  0.00005 

S  =  O.OOOI 

S  =  O.OI 

a.  =  0.025 

a  =  0.030 

a  =  0.025 

a  =  0.030 

a  =  0.025 

a  =  0.030 

0-5 

38 

31 

41 

33 

47 

37 

I. 

49 

40 

52 

42 

56 

45 

1-5 

57 

47 

59 

48 

62 

51 

2. 

64 

52 

65 

53 

67 

54 

3- 

72 

59 

72 

59 

72 

60 

4- 

77 

64 

77 

64 

76 

63 

5- 

Si 

68 

80 

68 

79 

66 

6. 

86 

72 

84 

71 

So 

68 

8. 

9i 

76 

87 

74 

82 

70 

10. 

96 

80 

9i 

80 

85 

73 

15- 

105 

89 

97 

84 

90 

77 

25- 

114 

100 

101 

92 

95 

82 

HO  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

here  given,  showing  their  average  values  for  three  slopes  and 
two  degrees  of  roughness. 

As  an  example  of  the  use  of  the  table  let  it  be  required  to 
find  the  width  and  depth  of  a  ditch  of  most  advantageous  cross- 
section,  whose  channel  is  to  be  in  tolerably  good  order,  so  that 
a  =  0.025.  The  amount  of  water  to  be  delivered  is  200  cubic 
feet  per  second  and  the  grade  is  I  in  1000,  the  side  slopes  of 
the  channel  being  I  to  I.  From  Art.  98  the  relation  between 
the  bottom  width  and  the  depth  of  the  water  is,  since  9  =  45°, 

b  =  d  \-^—n  —  2  cot  8}  =  o.82&£ 
\sm  0  I 

The  area  of  the  cross-section  then  is 

a  —  d(b-^d  cot  0)  =  1.828^, 
and  the  wetted  perimeter  is 


whence  the  hydraulic  radius  is  found  to  be 


It  is  indeed  a  general  rule,  which  might  properly  have  been 
set  forth  in  Art.  98,  that  the  hydraulic  radius  is  one-half  the 
depth  of  the  water  in  trapezoidal  channels  of  most  advanta-' 
geous  cross-section.  Now,  since  d  is  unknown  c  cannot  be 
taken  from  the  table,  and  as  a  first  approximation  let  it  be  sup- 
posed to  be  60.  Then  in  the  general  formula  for  discharge  the 
above  values  are  substituted,  giving 


200  =  60  x  1.828^'  Vo.$d  x  o.ooi, 

from  which  d  is  found  to  be  5.8  feet.  Accordingly  r  =  2.9 
feet,  and  from  the  table  c  is  about  71.  Repeating  the  compu- 
tation with  this  value  of  c  there  is  found  d=  5.44  feet,  which, 


ART.  103.]  DITCHES  AND   CANALS.  241 

considering  the  uncertainty  of  c,  is  sufficiently  close.  The 
depth  may  then  be  made  5.5  feet,  and  the  bottom  width  will 
be 

b  =  0.828  x  5-5  =  4-55  feet, 

and  the  sectional  area  is 

a  =  1.828  X  5-52  =  55.3  square  feet, 
which  gives  for  the  velocity 

200 

v  =  -  -  =  3.62  feet  per  second. 
55-3 

This  completes  the  investigation  if  the  velocity  is  regarded  as 
satisfactory.  But  for  most  earths  this  would  be  too  high,  and 
accordingly  the  section  must  be  made  wider  and  of  less  depth 
in  order  to  reduce  the  hydraulic  radius  and  diminish  the  ve- 
locity. 

The  following  statements  show  approximately  the  veloci- 
ties which  are  required  to  move  different  materials : 

0.25  feet  per  second  moves  fine  clay, 

0.5  feet  per  second  moves  loam  and  earth, 

i.o  feet  per  second  moves  sand, 

2.0  feet  per  second  moves  gravel, 

3.0  feet  per  second  moves  pebbles  I  inch  in  size, 

4.0  feet  per  second  moves  spalls  and  stones, 

6.0  feet  per  second  moves  large  stones. 

The  mean  velocity  in  a  channel  may  be  somewhat  larger  than 
these  values  before  the  materials  will  move,  because  the  veloci- 
ties along  the  wetted  perimeter  are  smaller  than  the  mean 
velocity.  More  will  be  found  on  this  subject  in  Art.  107, 

Prob.  129.  Compute  the  mean  velocity  in  a  ditch  which  is 
to  discharge  200  cubic  feet  per  second  on  a  grade  of  I  in  lOOO 
when  its  bottom  width  is  16  feet  and  the  side  slopes  are  i  to  I. 
Ans.  d  —  3.09  feet,  v  —  3.4  feet,  per  second. 


242  FLOW  IN  CONDUITS  AND    CANALS.       [CHAP.  VIII. 

ART.  104.  LOSSES  OF  HEAD. 

The  only  loss  of  head  thus  far  considered  is  that  due  to 
friction,  but  other  sources  of  loss  may  often  exist.  As  in  the 
flow  in  pipes,  these  may  be  classified  as  losses  at  entrance,  losses 
due  to  curvature,  and  losses  caused  by  obstructions  in  the  chan- 
nel or  by  changes  in  the  area  of  cross-section. 

When  water  is  admitted  to  a  channel  from  a  reservoir  or 
pond  through  a  rectangular  sluice  there  occurs  a  contraction 

similar  to  that  at  the  entrance 
into  a  pipe,  and  which  may  be 
often  observed  in  a  slight  de- 
pression of  the  surface,  as  at  D 
FIG.  71.  in  the  diagram.  At  this  point, 

therefore,  the  velocity  is  greater  than  the  mean  velocity  v,  and 
a  loss  of  energy  or  head  results  from  the  subsequent  expansion, 
which  is  approximately  measured  by  the  difference  of  the 
depths  d^  and  </2,  the  former  being  taken  at  the  entrance  of  the 
channel,  and  the  latter  below  the  depression  where  the  uniform 
flow  is  fully  established.  According  to  the  experiments  of 
DUBUAT,  the  loss  of  head  is  measured  by 


in  which  m  ranges  between  o  and  2  according  to  the  condition 
of  the  entrance.  If  the  channel  be  small  compared  with  the 
reservoir,  and  both  the  bottom  and  side  edges  of  the  entrance 
be  square,  m  may  be  nearly  2  ;  but  if  these  edges  be  rounded,  ?;/ 
may  be  very  small,  particularly  if  the  bottom  contraction  is 
suppressed.  All  the  remarks  in  Chapter  IV  relating  to  sup- 
pression of  the  contraction  apply  here,  and  in  a  short  channel 
or  flume  it  may  be  important  to  prevent  this  loss  of  head  by  a 
rounded  or  curved  approach. 


ART.  104.]  LOSSES  OF  HEAD.  243 

The  loss  of  head  due  to  bends  or  curves  in  the  channel  is 
small  if  the  curvature  be  slight.  Undoubtedly  every  curve 
offers  a  resistance  to  the  change  in  direction  of  the  velocity, 
and  thus  requires  an  additional  head  to  cause  the  flow  beyond 
that  needed  to  overcome  the  frictional  resistances.  Several 
formulas  have  been  proposed  to  express  this  loss,  but  they  all 
seem  unsatisfactory,  and  hence  will  not  be  presented  here,  par- 
ticularly as  the  data  for  determining  their  constants  are  very 
scant.  It  will  be  plain  that  the  loss  of  head  due  to  a  curve 
increases  with  its  length  and  decreases  with  its  radius.  Art. 
131  gives  a  discussion  concerning  the  cause  of  losses  in  bends 
and  curves. 

The  losses  of  head  caused  by  sudden  enlargement  or  by 
sudden  contraction  of  the  cross-section  of  a  channel  may  be 
estimated  by  the  rules  deduced  in  Arts.  68  and  69.  In  order 
to  avoid  these  losses  changes  of  section  should  be  made  grad- 
ually, so  that  energy  may  not  be  lost  in  impact.  Obstructions 
or  submerged  dams  may  be  regarded  as  causing  sudden  changes 
of  section,  and  the  accompanying  losses  of  head  are  governed 
by  similar  laws.  The  numerical  estimation  of  these  losses  will 
generally  be  difficult,  but  the  principles  which  control  them 
will  often  prove  useful  in  arranging  the  design  of  a  channel  so 
that  the  maximum  work  of  the  water  can  be  rendered  avail- 
able. But  as  all  losses  of  head  are  directly  proportional  to  the 

v* 

velocity-head  — ,  it  is  plain  that  they  can  be  rendered  inappre- 
ciable by  giving  to  the  channel  such  dimensions  as  will  render 
the  mean  velocity  very  small.  This  may  sometimes  be  impor- 
tant in  a  short  conduit  or  flume  which  conveys  water  from  a 
pond  or  reservoir  to  a  hydraulic  motor,  particularly  in  cases 
where  the  supply  is  scant,  and  where  all  the  available  head  is 
required  to  be  utilized. 

Prob.   130.  Explain  what  will  happen  when  in  a  channel 


244  FLOW  IN  CONDUITS  AND   CANALS.       [CHAP.  VIII. 

which  conveys  50  cubic  feet  of  water  per  second  the  cross-sec- 
tion suddenly  changes  from  5  to  25  square  feet. 


ARTICLE  105.  THE  ENERGY  OF  THE  FLOW. 

If  all  the  filaments  of  a  stream  of  water  flowing  in  a  pipe, 
conduit,  or  canal  have  the  same  uniform  velocity  vy  the  poten- 
tial energy  per  second  is  the  weight  W  of  the  discharge  per 

2 

second  multiplied  by  its  velocity-head  —  ;  or  if  a  be  the  cross- 

section  of  the  stream  and  w  the  weight  of  a  cubic  unit  of  water, 
the  energy  is 

V*  V*  V* 

K  =  W  —  =  wq  —  =  wa  —  , 

lg  *  *g  *g 

in  which  W  is  the  total  weight  of  water  delivered  per  second 
and  w  is  the  weight  of  one  cubic  foot.  In  this  case,  then,  the 
work  of  a  stream  of  constant  cross-section  varies  as  the  cube  of 
its  velocity. 

The  velocities  of  the  filaments  in  a  cross-section  are,  how- 
ever, not  uniform,  some  being  less  and  others  greater  than  the 
mean  velocity  v,  so  that  the  above  expression  for  K  does  not 
truly  represent  the  energy  of  the  discharge.  Let  the  cross- 
section  be  divided  into  a  number  n  of  elementary  sections,  each 
of  which  is  equal  to  a'  ;  then  the  mean  velocity  is 


and  the  true  energy  of  the  discharge  is 


2g  2g 

The  true  energy  K'  may  be  greater  ,  or  less  than  that  repre- 
sented by  K  according  to  the  manner  in  which  the  velocities 
vary  throughout  the  cross-section. 


ART.  105.] 


THE  ENERGY  OF   THE  FLOW. 


245 


For  this  purpose  let  the  equation  for  K'  be  divided  by  that 
K,  and  n  be  placed  for  the  ratio  a  -v-  a',  giving 


K 


nv* 


nv*  * 


Now  let  u  be  the  difference  between  the  mean  v  and  any  in- 
dividual velocity,  so  that  vl  =  v  ±  ult  vz  =  v  ±  #a,  etc.;  then 

^3  =  2if  ±  $2i?u  +  $2vu*  ±  2u\ 

But  it  is  a  property  of  the  arithmetical  mean  that  2u  =  o  ; 
hence  the  term  containing  u  disappears,  and  since  m>*  =  -2V, 
the  expression  becomes 


K 


3  2V  , 


nv 


(75) 


Therefore  K'  is  greater  or  less  than  K  according  as 
is  positive  or  negative. 

It  is  difficult,  if  not  impossible,  to  give  even  a  general  state- 
ment of  the  percentage  which  is  to  be  added  to  the  energy  K 
in  order  to  find  the  true  energy  K  '.  •  In  a  circular  pipe  or  small 


SCALE  OF  FEET 


FIG.  72. 


trough  the  velocities  may  not  greatly  differ,  so  that  K  and  K' 
may  closely  agree.     The  experiments  of  FTELEY  and  STEARNS 


246  FLOW  IN  CONDUITS  AND    CANALS.       [CHAP.  VIIL 

on  the  Sudbury  conduit  furnish  the  means  of  computing  this 
percentage  for  several  cases,  one  of  which  is  represented  in 
Fig.  72.*  This  shows  the  cross-section  of  the  conduit  when 
the  water  was  about  3  feet  deep,  the  dots  being  the  points  at 
which  the  velocities  were  measured  by  a  current  meter  (Art. 
109),  and  the  figures  giving  the  observed  values  in  feet  per 
second.  The  number  of  these  velocities  is  n  =  97,  and  their 
mean  is  ^=2.620.  By  cubing  the  individual  velocities  and 
comparing  their  sum  with  nv*,  or  by  the  use  of  formula  (75), 
there  is  found  K'  =  0.9992 K.  Hence  for  this  particular  case 
the  mean  and  true  energies  are  closely  equal. 

Prob.  131.  The  discharge  of  the  Sudbury  conduit  under  the 
conditions  above  described  was  64.43  cubic  feet  per  second. 
Compute  the  theoretic  horse-power  of  the  flow. 

*  Transactions  American  Society  o*  <3ivil  Engineers,  1883,  vol.  xii.  p.  324. 


ART.  io6.J  BROOKS  AND  RIVERS.  247 


CHAPTER   IX. 
FLOW   IN    RIVERS. 

ARTICLE  106.  BROOKS  AND  RIVERS. 

No  branch  of  Hydraulics  has  received  more  detailed  investi- 
gation than  that  of  the  flow  in  river  channels,  and  yet  the  sub- 
ject is  but  imperfectly  understood.  The  great  object  of  all 
these  investigations  has  been  to  devise  a  simple  method  of  de- 
termining the  mean  velocity  and  discharge  without  the  neces- 
sity of  expensive  field  operations.  In  general  it  may  be  said 
that  this  end  has  not  yet  been  attained,  even  for  the  case  of 
uniform  flow.  Of  the  various  formulas  proposed  to  represent 
the  relation  of  mean  velocity  to  the  hydraulic  radius  and  the 
slope,  none  have  proved  to  be  of  general  practical  value  except 
the  empirical  expression  used  in  the  last  chapter,  and  this  is 
often  inapplicable  on  account  of  the  difficulty  of  measuring  s 
and  determining  c.  The  fundamental  equations  for  discussing 
the  laws  of  variation  in  the  mean  velocity  v  and  in  the  dis- 
charge q  are 

•v  =  c  y rsy  q  =  a  .  c  Vrs~: 

and  all  the  general  principles  of  the  last  chapter  are  to  be  taken 
as  directly  applicable  to  uniform  flow  in  natural  channels. 

KUTTER'S  formula  for  the  value  of  c  is  probably  the  best 
in  the  present  state  of  science,  although  it  is  now  generally 
recognized  that  it  gives  too  large  values  for  small  slopes.  In 
using  it  the  coefficients  for  rivers  in  good  condition  may  be 
taken  from  Art.  103,  but  for  bad  regimen  a  is  to  be  taken  at  0.03, 
and  for  wild  torrents  at  0.04  or  0.05.  It  is,  however,  too  much  to 


248  FLOW  IN  RIVERS.  [CHAP.  IX. 

expect  that  a  single  formula  should  accurately  express  the 
mean  velocity  in  small  brooks  and  large  rivers,  and  the  general 
opinion  now  is  that  efforts  to  establish  such  an  expression  will 
not  prove  successful.  In  the  present  state  of  the  science  no 
engineer  can  afford  in  any  case  of  importance  to  rely  upon  a 
formula  to  furnish  anything  more  than  a  rough  approximation 
to  the  discharge  in  river  channels,  but  actual  field  measure- 
ments of  velocity  must  be  made. 

When  the  above  formulas  are  used  to  determine  the  dis- 
charge of  a  river  a  long  straight  portion  or  reach  should  be 
selected,  where  the  cross-sections  are  uniform  in  shape  and  size. 
The  width  of  the  stream  is  then  divided  into  a  number  of  parts, 
and  soundings  taken  at  each  point  of  division.  The  data  are 
thus  obtained  for  computing  the  area  a  and  the  wetted  perime- 
ter/, frqm  which  the  hydraulic  depths  is  derived.  To  deter- 
mine the  slope  s  a  length  /  is  to  be  measured,  at  each  end  of 
which  bench  marks  are  established  whose  difference  of  elevation 
is  found  by  precise  levels.  The  elevations  of  the  water  surfaces 
below  these  benches  are  then  to  be  simultaneously  taken, 
whence  the  fall  //  in  the  distance  /  becomes  known.  As  this 
fall  is  often  small,  it  is  very  important  that  every  precaution  be 
taken  to  avoid  error  in  the  measurements,  and  that  a  number 
of  them  be  taken  in  order  to  secure  a  precise  mean.  Care 
should  be  observed  that  the  stage  of  water  is  not  varying  while 
these  observations  are  being  made,  and  for  this  and  other  pur- 
poses a  permanent  gauge  must  be  established.  It  is  also  very 
important  that  the  points  upon  the  water  surface  which  are 
selected  for  comparison  should  be  situated  so  as  to  be  free  from 
local  influences  such  as  eddies,  since  these  often  cause  marked 
deviations  from  the  normal  surface  of  the  stream.  If  hook 
gauges  can  be  used  for  referring  the  water  levels  to  the  benches 
probably  the  most  accurate  results  can  be  obtained.  It  has 
been  observed  that  the  surface  of  a  swiftly  flowing  stream  is 


ART.  107.]  VELOCITIES  IN  A    CROSS-SECTION.  249 

not,  a  plane,  but  a  cylinder,  which  is  concave  to  the  bed,  its 
highest  elevation  being  where  the  velocity  is  greatest,  and 
hence  the  two  points  of  reference  should  be  located  similarly 
with  respect  to  the  axis  of  the  current.  In  spite  of  all  precau- 
tions, however,  the  relative  error  in  h  will  usually  be  large  in 
the  case  of  slight  slopes,  unless  /  be  very  long,  which  cannot 
often  occur  in  streams  under  conditions  of  uniformity. 

Owing  to  the  uncertainty  of  determinations  of  discharge 
made  in  the  manner  just  described,  the  common  practice  is  to 
gauge  the  stream  by  velocity  observations,  to  which  subject 
therefore  a  large  part  of  this  chapter  will  be  devoted.  The 
methods  given  are  equally  applicable  to  conduits  and  canals, 
and  in  Art.  115  will  be  found  a  summary  which  briefly  com- 
pares the  various  processes. 

Prob.  132.  Which  has  the  greater  discharge — a  stream  2 
feet  deep  and  85  feet  wide  on  a  slope  of  I  foot  per  mile,  or  a 
stream  3  feet  deep  and  40  feet  wide  on  a  slope  of  2  feet  per 
mile  ? 


ARTICLE  107.  VELOCITIES  IN  A  CROSS-SECTION. 

The  mean  velocity  v  is  the  average  of  all  the  velocities  of 
all  the  small  sections  or  filaments  in  a  cross-section  (Art.  93). 
Some  of  these  individual  velocities  are  much  smaller,  and  oth- 
ers materially  larger,  than  the  mean  velocity.  Along  the  bot- 
tom of  the  stream,  where  the  frictional  resistances  are  the  great- 
est, the  velocities  are  the  least ;  along  the  centre  of  the  stream 
they  are  the  greatest.  A  brief  statement  of  the  general  laws 
of  variation  of  these  velocities  is  now  to  be  made. 

In  Fig.  73  there  is  shown  at  A  a  cross-section  of  a  stream 
with  contour  curves  of  equal  velocity ;  here  the  greatest  veloc- 
ity is  seen  to  be  near  the  deepest  part  of  the  section  a  short 


250  FLOW  IN  RIVERS.  [CHAP.  IX. 

distance  below  the  surface.  At  B  is  shown  a  plan  of  «£he 
stream  with  arrows  roughly  representing  the  intensities  of  the 
surface  velocities  at  different  points ;  the  greatest  of  these  is 
seen  to  be  near  the  deepest  part  or  axis  of  the  channel  while 
the  others  diminish  toward  the  banks,  the  law  of  variation  be- 
ing a  curve  resembling  a  parabola.  At  C  is  shown  by  arrows 
the  variation  of  velocities  in  a  vertical  line,  the  smallest  being 


FIG.  73. 

at  the  bottom,  and  the  largest  a  short  distance  below  the  sur- 
face ;  concerning  this  curve  there  has  been  much  contention, 
but  it  is  commonly  thought  to  be  a  parabola  whose  axis  is  hori- 
zontal. These  are  the  general  laws  of  the  variation  of  velocity 
throughout  the  cross-section  ;  the  particular  relations  are  of  a 
complex  character,  and  vary  so  greatly  in  channels  of  different 
kinds  that  it  is  difficult  to  formulate  them,  although  many  at- 
tempts to  do  so  have  been  made.  Some  of  these  formulas 
which  connect  the  mean  velocity  with  particular  velocities,  such 
as  the  maximum  surface  velocity,  mid  depth  velocity  in  the 
axis  of  the  stream,  etc.,  will  be  given  in  the  following  articles 
in  connection  with  the  subject  of  gauging  rivers. 

In  a  straight  channel  whose  bed  is  of  a  uniform  nature  the 
deepest  part  is  near  the  middle  of  its  width,  while  the  two  sides 
are  approximately  symmetrical.  In  a  river  bend,  however, 
the  deepest  part  is  near  the  outer  bank,  while  on  the  inner 
side  the  water  is  shallow :  the  cause  of  this  is  undoubtedly  due 
to  the  centrifugal  force  of  the  current,  which,  resisting  the 


ART.  io8.]      TRANSPORTING  CAPACITY  OF  CURRENTS.  251 

change  in  direction,  creates  currents  which  scour  away  the 
outer  bank  or  prevents  deposits  from  there  occurring.  It  is 
well  known  to  all,  that  rivers  of  the  least  slope  have  the  most 
bends ;  perhaps  this  is  due  to  the  greater  relative  influence  of 

such  cross  currents  (Art.  131). 

i 

The  theory  of  the  flow  of  water  in  channels,  like  that  of 
flow  in  pipes,  is  based  upon  the  supposition  of  a  mean  velocity 
which  is  the  average  of  all  the  parallel  individual  velocities  in 
the  cross-section.  But  in  fact  there  are  numerous  sinuous 

•     • 

motions  of  particles  from  the  bottom  to  the  surface  which  also 
consume  a  portion  of  the  lost  head.  The  influence  of  these 
sinuosities  is  as  yet  but  little  understood ;  when  in  the  future 
this  becomes  known  a  better  theory  may  be  possible. 

Prob.  133.  Find  the  approximate  discharge  of  a  stream 
whose  width  is  200  feet,  depth  3  feet,  slope  0.6  feet  per  mile, 
when  the  bottom  is  very  stony  and  in  bad  condition. 

ARTICLE  108.  THE  TRANSPORTING  CAPACITY  OF  CURRENTS. 

The  fact  that  the  water  of  streams  transports  large  quan- 
tities of  earthy  matter,  either  in  suspension  or  by  rolling  it 
along  the  bed  of  the  channel,  is  well  known,  and  has  already 
been  mentioned  in  Article  103.  It  is  now  to  be  shown  that 
the  diameters  of  bodies  which  can  be  moved  by  the  pressure 
of  a  current  vary  as  the  square  of  its  velocity,  and  their  weights 
vary  as  the  sixth  power  of  the  velocity. 

When  water  causes  sand  or  pebbles  to  roll  along  the  bed 
of  a  channel  it  must  exert  a  force  approximately  proportional 
to  the  square  of  the  velocity  and  to  the  area  exposed  (Art.  32), 
or  if  d  be  the  diameter  of  the  body  and  a  a  constant, 

F  =  ad*v\ 

But  if  motion  just  occurs,  this  force  is  also  proportional  to  the 
weight  of  the  body,  because  the  frictional  resistances  of  one 


252  FLOW  IN  RIVERS.  [CHAP.  IX. 

body  upon  another  varies  as  the  normal  pressure  or  weight. 
And  as  the  weight  varies  as  the  cube  of  the  diameter, 

d*  =  ad*v\         or         d=av\ 

Now  since  d  varies  as  v*y  the  weight  of  the  body,  which  is  pro- 
portional to  d3,  must  vary  as  v* ;  which  proves  the  proposition 
as  enunciated. 

Since  the  weight  of  sand  and  stones  when  immersed  in 
water  is  only  about  one-half  their  weight  in  air,  the  frictional 
resistances  to  their  motion  are  slight,  and  this  helps  to  explain 
the  circumstance  that  they  are  so  easily  transported  by  currents 
of  moderate  velocity.  It  is  found  by  observation  that  a  pebble 
about  one  inch  in  diameter  is  rolled  along  the  bed  of  a  channel 
when  the  velocity  is  about  3^  feet  per  second  ;  hence,  according 
to  the  above  theoretical  deduction,  a  velocity  5  times  as  great,, 
or  17^-  feet  per  second,  will  carry  along  stones  of  25  inches 
diameter.  This  law  of  the  transporting  capacity  of  flowing 
water  is  only  an  approximate  one,  for  the  recorded  experiments 
seem  to  indicate  that  the  diameters  of  moving  pebbles  on  the 
bed  of  a  channel  do  not  vary  quite  as  rapidly  as  the  square  of 
the  velocity.  The  law,  moreover,  is  applicable  only  to  bodies 
of  similar  shape,  and  cannot  be  used  for  comparing  round  peb- 
bles with  flat  spalls. 

The  following  table  gives  the  velocities  on  the  bed  or  bot- 
tom of  the  channel  which  are  required  to  move  the  materials 
stated.  The  corresponding  approximate  mean  velocities  in 
the  cross-section  given  in  the  last  column  are  derived  from  the 
empirical  formula  deduced  by  DARCY, 

v  =  vf  +  1 1  Vrs, 

in  which  v'  is  the  bottom  and  v  the  mean  velocity.  The  bot- 
tom or  transporting  velocities  were  deduced  by  DUBUAT  from 
experiments  in  small  troughs,  and  hence  are  probably  slightlj 


ART.  109.]  THE   CURRENT  METER.  253 

less  than  the  velocities  which  would  move  the  same  mate- 
rials in  channels  of  natural  earth. 

Bottom          Mean 
velocity.       velocity. 

Clay,  fit  for  pottery, 0.3  0.4 

Sand,  size  of  anise-seed, 0.4  0.5 

Gravel,  size  of  peas, 0.6  0.8 

Gravel,  size  of  beans, 1.2  1.6 

Shingle,  about  I  inch  in  diameter,    ...  2.5  3.5 

Angular  stones,  about  i£  inches  diameter,  3.5  4.5 

The  general  conclusion  to  be  derived  from  these  figures  is 
that  ordinary  small,  loose  earthy  materials  will  be  transported 
or  rolled  along  the  bed  of  a  channel  by  velocities  of  2  or  3  feet 
per  second.  It  is  not  necessarily  to  be  inferred  that  this 
movement  of  the  materials  is  of  an  injurious  nature  in  streams 
with  a  fixed  regimen,  but  in  artificial  canals  the  subject  is  one 
that  demands  close  attention.  The  velocity  of  the  moving 
objects  after  starting  has  been  found  to  be  usually  less  than 
half  that  of  the  current.* 

Prob.  134.  A  stone  weighing  0.5  pounds  is  moved  by  a 
current  of  3  feet  per  second ;  what  weight  will  be  moved  by  a 
current  of  9  feet  per  second  ? 

ARTICLE  109.  THE  CURRENT  METER. 

The  most  convenient  way  of  precisely  measuring  the  dis- 
charge of  a  canal,  conduit,  or  small  stream  is  by  means  of  a 
weir  which  is  specially  built  for  that  purpose.  The  flow  of  a 
very  large  conduit  or  of  a  large  stream  cannot,  however,  be 
successfully  gauged  in  this  manner,  both  on  account  of  the 
expense  of  the  dam  and  weir,  and  because  the  weir  coefficients 
are  not  well  known  for  depths  of  water  greater  than  about  1.5 

*  See  paper  by  HERSCHEL  on  the  erosive  and  abrading  power  of  water,  in 
Journal  of  the  Franklin  Institute,  May,  1878. 


254  FLOW  IN  RIVERS.  [CHAP.  IX. 

feet.  Large  quantities  of  water,  therefore,  are  usually  meas- 
ured by  observing  the  velocity  of  its  flow,  and  the  current 
meter  furnishes  a  method  of  doing  this  which  is  extensively 
used,  and  which  gives  accurate  results. 

The  current  meter  is  like  a  windmill,  having  three  or  more 
vanes  mounted  on  a  spindle,  and  so  arranged  that  the  face  of 
the  mill  or  wheel  always  stands  normal  to  the  current,  the 
pressure  of  which  causes  it  to  revolve.  The  number  of  revo- 
lutions of  the  wheel  is  approximately  proportional  to  the  ve- 
locity of  the  current.  In  the  best  forms  of  instruments  the 
number  of  revolutions  made  in  a  given  time  is  determined  by 
an  apparatus  on  shore  or  in  a  boat  from  which  wires  lead  to 
the  meter  under  water ;  at  every  revolution  an  electric  connec- 
tion is  made  and  broken  which  affects  a  dial  on  the  recording 
apparatus.  The  observer  has  hence  only  to  note  the  time  of 
beginning  and  ending  of  the  experiment,  and  to  read  the  num- 
ber of  revolutions  which  have  occurred  during  the  interval. 
For  a  canal  or  small  stream  the  meter  is  best  operated  from  a 
bridge  ;  in  large  streams  a  boat  must  be  used. 

To  derive  the  velocity  from  the  number  of  recorded  revo- 
lutions per  second,  the  meter  must  be  first  rated  by  pushing  it 
at  a  known  velocity  in  still  water.  For  this  purpose  a  base 
line  several  hundred  feet  long  is  laid  out  on  shore,  and  ranges 
established  so  that  a  boat  may  be  rowed  over  the  same  dis- 
tance and  the  time  of  its  passage  be  determined.  The  current 
meter  is  placed  in  the  bow  of  the  boat,  and  a  start  made  suffi- 
ciently far  from  the  base  so  that  a  uniform  velocity  can  be 
acquired  before  reaching  it;  the  distance  is  then  traversed 
with  this  uniform  velocity  and  the  times  observed,  as  also  the 
actual  records  of  the  meter.  It  is  usually  found  that  the  num- 
ber of  revolutions  are  not  exactly  proportional  to  the  actual 
velocities  of  the  boat,  and  hence  it  is  necessary  to  run  the  boat 


ART.  109.]  THE   CURRENT  METER.  2$$ 

at  different  velocities  per  second  and  ascertain  the  correspond- 
ing number  of  revolutions  of  the  wheel  for  each.  A  table  may 
then  be  prepared  which  gives  the  velocity  corresponding  to 
the  revolutions  per  second,  from  which  in  subsequent  field 
work  the  reductions  can  readily  be  made.  The  relation  be- 
tween the  velocity  V  and  the  number  of  revolutions  per 
second  n  can  also  be  expressed  by  an  equation  of  the  form 


and  the  experiments  furnish  the  data  from  which  the  coeffi- 
cients a,  ft,  and  y  can  be  determined  by  the  help  of  the  Method 
of  Least  Squares.  For  ordinary  ranges  of  velocity  y  is  usually 
a  small  quantity,  and  it  is  often  taken  as  zero. 

A  current  meter  cannot  be  used  for  determining  the  ve- 
locity in  a  small  trough,  for  the  introduction  of  it  into  the 
cross-section  would  contract  the  area  and  cause  a  change  in  the 
velocity  in  front  of  the  wheel.  In  large  conduits,  canals,  and 
rivers  it  is,  however,  one  of  the  most  convenient  and  accurate 
instruments.  By  holding  it  at  a  fixed  position  below  the  sur- 
face the  velocity  at  that  point  is  found  ;  by  causing  it  to  de- 
scend at  a  uniform  rate  from  surface  to  bottom  the  mean  ve- 
locity in  that  vertical  is  obtained  ;  and  by  passing  it  at  a  uni- 
form rate  over  all  parts  of  the  cross-section  of  a  channel  the 
mean  velocity  v  is  directly  determined.  It  is  usually  mounted 
at  the  end  of  a  long  pole,  which  is  graduated  so  that  the  depth 
of  the  meter  below  the  water  surface  can  be  directly  read.* 

Prob.  135.  By  rating  a  certain  water  meter,  the  equation 
F=o.i59+  i  905;^  was  deduced  for  velocities  varying  from 
i  to  7  feet  per  second.  Compute  the  velocity  of  the  current 
when  the  wheel  revolves  101  times  in  41  seconds. 

*See  paper  by  STEARNS  in  Transactions  American  Society  of  Civil  En- 
gineers, 1883,  vol.  xii.  p.  301,  for  detailed  account  of  the  use  of  the  current 
meter  in  the  Sudbury  conduit. 


256  FLOW  IN  RIVERS.  [CHAP.  IX. 

ARTICLE  no.  FLOATS. 

The  method  for  measuring  the  discharge  of  streams  which 
has  been  most  extensively  used  is  by  observing  the  velocity  of 
flow  by  the  help  of  floats.  Of  these  there  are  three  kinds,  sur- 
face floats,  double  floats,  and  rod  floats.  Surface  floats  should 
be  sufficiently  submerged  so  as  to  thoroughly  partake  of  the 
motion  of  the  upper  filaments,  and  should  be  made  of  such  a 
form  as  not  to  be  readily  affected  by  the  wind.  The  time  of 
their  passage  over  a  given  distance  is  determined  by  two  ob- 
servers at  the  ends  of  a  base  on  shore  by  stop-watches ;  or  only 
one  watch  may  be  used,  the  instant  of  passing  each  section 
being  signalled  to  the  time-keeper.  If  /  be  the  length  of  the 
base,  and  t  the  time  of  passage  in  seconds,  the  velocity  of  the 
float  per  second  is 


The  numerical  work  of  division  can  here,  as  in  other  cases,  be 
best  performed  by  taking  the  reciprocal  of  t  from  a  table,  and 
multiplying  it  by  /,  which  for  convenience  may  be  an  even 
number,  such  as  100  or  200  feet. 

A  sub-surface  float  consists  of  a  small  surface  float  con- 
nected by  a  fine  cord  or  wire  with  the  large  real  float  which  is 
weighted  so  as  to  remain  submerged,  and  keep  the  cord  rea- 
sonably taut.  The  surface  float  should  be  made  of  such  a 
form  as  to  offer  but  slight  resistance  to  the  motion,  while  the 
lower  float  is  large,  it  being  the  object  of  the  combination  to 
determine  the  velocity  of  the  lower  one  alone.  This  arrange- 
ment has  been  extensively  used,  but  it  is  probable  that  in  all 
cases  the  velocity  of  the  large  float  is  somewhat  affected  by 
that  of  the  upper  one,  as  well  as  by  the  friction  of  the  cord. 
In  general  the  use  of  these  floats  is  not  to  be  encouraged,  if 
any  other  method  of  measurement  can  be  devised. 


ART.  no.]  FLOATS. 

The  rod  float  is  a  hollow  cylinder  of  tin,  which  can  be 
weighted  by  dropping  in  pebbles  or  shot  so  as  to  stand  vertical- 
ly at  any  depth.  When  used  for  Velocity  determinations  they 
are  weighted  so  as  to  reach  nearly  to  the  bottom  of  the  chan- 
nel, and  the  time  of  passage  over  a  known  distance  determined 
as  above  explained.  It  is  often  stated  that  -the  velocity  of  a 
rod  float  -is  the  mean  velocity  of  all  the  filaments  in  the  verti- 
cal plane  in  which  it  moves.  Theoretically  this  is  not  the  case  ; 
and  experiments  by  FRANCIS  have  proved  that  the  velocity  of- 
the  rod  is  usually  from  i  to  5  per  cent  less  than;  that  of  the 
mean  velocity  in  the  vertical.  FRANCIS  has  also  deduced  the 
following  empirical  formula  for  finding  the  mean  velocity  Vm 
from  the  observed  velocity  Vr  of  the  rod, 


—  Vr(  1.012-0.116  A/  -j-Y 


in  which  d  is  the  total  depth  of  the  stream,  and  d'  the  depth  of 
water  below  the  bottom  of  the  rod.*  This  expression  is  prob- 
ably not  a  valid  one,  unless  d'  is  less  than  about  one-quarter  of 
d\  usually  it  will  be  best  to  have  d'  as  small  as  the  character  of 
the  bed  of  the  channel  will  allow. 

- 

The  log  used  by  seamen  for  ascertaining  the  speed  of  ves- 
sels may  be  often  conveniently  used  as  a  surface  float  when 
rough  determinations  only  are  desired,  it  being  thrown  from  a 
boat  or  bridge.  The  cord  of  course  must  be  previously 
stretched  when  wet,  so  that  its  length  may  not  be  altered  by 
the  immersion  ;  if  graduated  by  tags  or  knots  in  divisions  of 
six  feet,  the  log  may  be  allowed  to  float  for  one  minute,  and 
then  the  number  of  divisions  run  out  in  this  time  will  be  ten 
times  the  velocity  in  feet  per  second. 

The  determination  of  particular  velocities  in  streams  by 
means  of  floats  appears  to  be  simple,  but  in  practice  many 

*  Lowell  Hydraulic  Experiments,  4th  Edition,  p.  195. 


258  FLOW  IN  RIVERS.  [CHAP.  IX. 

uncertainties  are  found  to  arise,  owing  to  wind,  eddies,  local 
currents,  etc.,  so  that  a  number  of  observations  are  generally 
required  to  obtain  a  precise  ftiean  result.  For  conduits,  canals, 
and  for  many  rivers  the  use  of  a  current  meter  will  be  found 
to  be  more  satisfactory  and  less  expensive  if  many  observa- 
tions are  required. 

Prob.  136.  A  rod  float  runs  a  distance  of  100  feet  in  42  sec- 
onds, the  depth  of  the  stream  being  6  feet,  while  the  foot  of 
the  rod  is  6  inches  above  the  bottom.  Compute  the  mean 
velocity  in  the  vertical. 

ARTICLE  in.  OTHER  CURRENT  INDICATORS. 

PlTOT's  tube  is  an  instrument  for  measuring  the  velocity  of 
a  current  by  the  velocity-head  which  it  will  produce.  In  its 

simplest  form  it  consists  of  a  bent 
glass  tube  as  shown  in  Fig.  74,  in 
which  the  mouth  of  the  submerged 
part  is  placed  so  as  to  directly  face 

the  current.     The  water  then  rises 

FIG.  74.  in  the  vertical  part  to  a  distance  k 

above  the  surface  of  the  flowing  stream,  and  the  velocity  is  ap- 
proximately equal  to  V2gh.  The  only  advantage  of  this  in- 
strument is  that  no  time  observation  is  necessary ;  the  disad- 
vantages are  many,  the  chief  being  that  the  distance  h  is  always 
very  small,  so  that  errors  are  liable  to  be  made  in  determining 
its  value.  As  actually  constructed,  PlTOT's  apparatus  generally 
consists  of  two  tubes  placed  side  by  side  with  their  submerged 
mouths  at  right  angles,  so  that  when  one  is  opposed  to  the  cur- 
rent, as  seen  in  Fig.  74,  the  other  stands  normal  to  it,  and  the 
water  surface  in  the  latter  tube  hence  is  at  the  same  level  as 
that  of  the  stream.  Both  tubes  are  provided  with  cocks  which 
may  be  closed  while  the  instrument  is  immersed,  and  it  can  be 
then  lifted  from  the  water  and  the  head  h  be  read  at  leisure. 


ART.  in.]  OTHER   CURRENT  INDICATORS. 


It  is  found  that  the  actual  velocity  is  always  less  than  VZgKt 
and  that  a  coefficient  must  be  deduced  for  each  instrument  by 
moving  it  in  still  water  at  known  velocities.  PlTOT's  tube  has 
been  but  little  used,  and  is  generally  regarded  as  an  imperfect 
instrument  for  velocity  determinations. 

The  hydrometric  pendulum,  shown  also  in  Fig.  74,  consists 
of  a  ball  suspended  from  a  string,  which  by  the  pressure  of  the 
current  is  kept  at  a  certain  inclination  from  the  vertical,  the 
angle  of  inclination  being  read  on  a  graduated  arc.  The  rela- 
tion between  this  angle  and  the  velocity  of  the  current  must 
be  determined  experimentally  before  the  instrument  can  be  used 
in  actual  observations.  This  apparatus  was  employed  by  some 
of  the  early  experimenters,  but  has  now  gone  out  of  use. 

The  hydrometric  balance  is  similar  in  principle  to  the  pen- 
dulum, the  string  being  replaced  by  a  rigid  rod  which  is  con- 
nected with  a  lever  at  its  upper  end,  upon  which  weights  are 
hung  so  as  to  keep  the  rod  in  a  vertical  position.  The  weights 
measure  the  intensity  of  the  pressure  of  the  current,  and  hence 
its  velocity,  the  relation  between  them  being  first  experimentally 
established  for  each  instrument.  The  hydrometric  balance  is  a 
mere  curiosity,  and  has  never  been  practically  used  for  velocity 
determinations.  A  torsion  balance,  in  which  the  pressure  of 
the  current  on  a  submerged  plate  causes  a  spring  to  be  tight- 
ened, has  also  been  devised.  All  the  instruments  mentioned 
in  this  article  are  adapted  only  to  the  measurement  of  veloci- 
ties in  small  troughs  or  channels,  and  even  for  these  have  been 
but  little  used. 

Prob.  137.  If  the  head  h  in  a  PlTOT  tube  is  o.oi  feet,  what 
is  the  approximate  velocity  of  the  current  ?  If  an  error  of  25 
per  cent  be  made  in  reading  /i,  how  does  this  affect  the  deduced 
value  of  the  velocity? 


260  FLO  W  IN  RIVERS.  [CHAP.  IX. 

ARTICLE  112.  GAUGING  THE  FLOW. 

The  most  common  method  of  gauging  the  flow  of  a  stream 
which  is  too  large  to  be  measured  by  a  weir  will  now  be  ex- 
plained. It  involves  field  operations  which,  although  simple 

in  statement,  generally  re- 
quire considerable  care  and 
expense.  In  all  cases  the 
first  step  should  be  to  estab- 
lish a  water  gauge  whose 

zero  is  located  with  reference  to  a  permanent  bench  mark,  so 
that  the  stage  of  water  at  any  time  may  be  determined.  Such 
a  gauge  is  usually  graduated  to  tenths  of  feet,  intermediate 
values  being  estimated  to  hundredths. 

One  or  more  sections  at  right  angles  to  the  direction  of  the 
current  are  to  be  established,  and  soundings  taken  at  intervals 
across  the  stream  upon  them,  the  water  gauge  being  read  while 
this  is  done.  The  distances  between  the  places  of  sounding 
are  measured  either  upon  a  cord  stretched  across  the  stream,  or 
by  other  methods  known  to  surveyors.  The  data  are  thus  ob- 
tained for  obtaining  the  areas  alt  a^,  #3,  etc.,  shown  upon  Fig. 
75,  and  the  sum  of  these  is  the  total  area  a.  Levels  should  be 
run  out  upon  the  bank  beyond  the  water's  edge,  so  that  in  case 
of  a  rise  of  the  stream  the  additional  areas  can  be  deduced. 
If  a  current  meter  is  used,  but  one  section  is  needed  ;  if  floats 
are  used,  at  least  two  are  required,  and  these  must  be  located  at 
a  place  where  the  channel  is  of  as  uniform  size  as  possible. 

The  mean  velocities  v1 ,  v^ ,  ^3 ,  etc.,  in  each  of  the  sections 
are  next  to  be  determined  for  each  of  the  sub-areas.  If  a  cur- 
rent meter  is  used,  this  may  be  done  by  starting  at  one  side  of  a 
subdivision,  and  lowering  it  at  a  uniform  rate  until  the  bottom 
is  nearly  reached,  then  moving  it  a  foot  or  two  horizontally  and 
raising  it  to  the  surface,  and  continuing  until  the  area  has  been 
covered.  The  velocity  then  deduced  from  the  whole  number 


ART.  112.]  GAUGING    THE  FLOW.  26 1 

of  revolutions  is  the  mean  velocity  for  the  subdivision.  Or  the 
meter  may  be  simply  raised  and  lowered  in  a  vertical  at  the 
middle  of  the  sub-area,  and  the  result  will  be  a  close  approxi- 
mation to  the  mean  velocity.  If  rod  floats  are  used  they  are 
started  above  the  upper  section,  and  the  times  of  passing  to  the 
lower  one  noted,  as  explained  in  Art.  no,  the  velocity  deduced 
from  afloat  at  the  middle  of  a  sub-area  being  taken  as  the 
mean  for  that  area.  It  will  be  found  that  the  rod  floats  are 
more  or  less  affected  by  wind,  whose  direction  and  intensity 
should  hence  always  be  noted. 

The  discharge  of  the  stream  is  the  sum  of  the  discharges 
through  the  several  sub-areas,  or 

q  =  a,v,  +  atvt  +  a,v,  +  etc.  ; 

and  if  this  be  divided  by  the  total  area  a,  the  mean  velocity 
for  the  entire  section  is  determined. 

The  following  notes  give  the  details  of  a  gauging  of  the 
Lehigh  River,  near  Bethlehem,  Pa.,  made  Oct.  15,  1885,  in  the 
above  manner  by  the  use  of  rod  floats.*  The  two  sections 
were  100  feet  apart,  divided  into  10  equal  divisions,  each  30  feet 
in  width,  except  the  one  at  the  north  bank,  which  was  32  feet. 
In  the  second  column  are  given  the  soundings  in  feet,  in  the 


Subdivisions.    Depths. 

Areas. 

Times. 

Velocities. 

Discharges. 

0.0 

55-5 

380 

0.263 

I4.6 

6° 

148.5 

220 

0  454 

67.4 

3                 '° 

201.7 

185 

0.540 

108.9 

4 

217-5 

120 

0.833 

181.2 

5 

210.0 

145 

0.690 

144.9 

6 

186.0 

150 

0.667 

124.  1 

7 

150.8 

165 

0.606 

91.4 

8              4'3 

114.0 

200 

0.500 

57-o 

9                  2.'2 

84.0 

320 

0.313 

26.3 

10 

o.o 

42.0 

430 

0.233 

9.8 

a  = 

1410.0 

9 

=  825.6 

*  Journal  of  Engineering  Society  of  Lehigh  University,  1885,  vol.  i.  p.  75. 


262  FLO  W  IN'  RIVERS,  [CHAP.  IX. 

third  the  areas  in  square  feet,  in  the  fourth  the  times  of  passage 
of  the  floats  in  seconds,  in  the  fifth  the  velocities  in  feet  per 
second,  which  are  directly  deduced  from  the  times  without  ap- 
plying the  correction  indicated  in  Art.  110,  and  in  the  last  are 
the  products  alvl ,  azv^ ,  which  are  the  discharges  for  the  sub- 
divisions at ,  az ,  etc.  The  total  discharge  is  found  to  be  826 
cubic  feet  per  second,  and  the  mean  velocity  is 

826 

v  = —  0.59  feet  per  second. 

1410 

A  second  gauging  of  the  stream,  made  a  week  later,  when  the 
water  level  was  0.59  feet  higher,  gave  for  the  discharge  1336 
cubic  feet  per  second,  for  the  total  area  1630  square  feet,  and 
for  the  mean  velocity  0.82  feet  per  second.  These  results  for 
discharge  and  velocity  should  probably  be  increased  about  3 
per  cent,  in  order  to  allow  for  the  difference  between  the  ve- 
locities as  observed  by  the  rod  floats,  and  the  true  mean  veloci- 
ties in  the  middle  of  the  sub-areas. 

As  to  the  accuracy  of  the  above  method,  it  may  be  said 
that  with  ordinary  work,  using  rod  floats,  the  discrepancies  in 
results  obtained  under  different  conditions  ought  not  to  exceed 
10  per  cent ;  and  with  careful  work,  using  current  meters,  they 
may  often  be  of  a  much  higher  degree  of  precision.  In  any 
event  the  results  derived  from  such  gaugings  of  rivers  are  more 
reliable  than  can  be  obtained  by  any  other  method. 

Prob.  138.  Compute  the  mean  depth  and  the  hydraulic 
radius  for  the  above  section  of  the  Lehigh  River. 

ARTICLE  113.  GAUGING  BY  SURFACE  VELOCITIES. 

If  by  any  means  the  mean  velocity  v  of  a  stream  can  be 
found,  the  discharge  is  known  from  the  relation  q  =  av,  the 
area  a  >being  measured  as  explained  in  the  last  article.  An  ap- 
proximate value  of  v  may  be  ascertained  by  one  or  more  float 


ART.  113.  ]          GAUGING  BY  SURFACE    VELOCITIES.  263 

measurements  by  means  of  t^e  known  relations  between  it  and 
the  surface  velocities. 

The  ratio  of  the  mean  velocity  v  to  the  maximum  surface 
velocity  Khas  been  found  to  usually  lie  between  0.7  and  0.85, 
and  about  o.S  appears  to  be  a  rough  mean  value.  Accordingly, 

v  =  o.8F; 

from  which,  if  V  be  accurately  determined,  v  can  be  computed 
with  an  uncertainty  usually  less  than  20  per  cent. 

Many  attempts  have  been  made  to  deduce  a  more  reliable 
relation  between  v  and  V.  The  following  rule  derived  from 
the  investigations  of  BAZIN  makes  the  relation  dependent  on 
the  coefficient  c,  whose  value  for  the  particular  stream  is  to  be 
obtained  from  the  evidence  presented  in  the  last  chapter  : 

V 


It  is  probable,  however,  that  the  relation  depends  more  on  the 
hydraulic  radius  and  the  shape  of  the  section  than  upon  the 
degree  of  roughness  of  the  channel,  which  c  mainly  represents. 

The  ratio  of  the  mean  velocity  vv  in  any  vertical  to  its  sur- 
face velocity  Vl  is  less  variable,  lying  between  0.85  and  0.92,  so 
that 


may  be  used  with  but  an  uncertainty  of  a  few  per  cent.  If 
several  velocities  V^  ,  F2  ,  etc.,  be  determined  by  surface  floats, 
the  mean  velocities  z\  ,  v9  ,  etc.,  for  the  several  sub-areas  a^  ,  a^  , 
etc.,  are  known,  and  the  discharge  is  q  —  alvl  -\-  a^  -f-  etc., 
as  before  explained.  This  method  will  usually  prove  unsatis- 
factory as  compared  with  the  use  of  rod  floats. 

Since  the  maximum   surface   velocity  is  greater  than  the 
mean  velocity  v,  and  since  the  velocities  at  the  shores  are 


264  FLOW  IN  RIVERS.  [CHAP.  IX. 

usually  very  small,  it  follows  that  there  are  in  the  surface  two 
points  at  which  the  velocity  is  equal  to  v.  If  by  any  means 
the  location  .of  either  of  these  could  be  discovered,  a  single 
velocity  observation  would  give  directly  the  value  of  v.  The 
position  of  these  points  is  subject  to  so  much  variation  in 
channels  of  different  forms,  that  no  satisfactory  method  of  lo- 
cating them  has  yet  been  devised. 

The  influence  of  wind  upon  the  surface  velocities  is  so  great, 
that  these  methods  of  determining  v  will  prove  useless,  except 
in  calm  weather.  A  wind  blowing  up  stream  decreases  the  sur- 
face velocities,  and  one  blowing  down  stream  increases  them, 
without  materially  affecting  the  mean  velocity  and  discharge. 

Prob.  139.  A  stream  60  feet  wide  is  divided  into  three  sec- 
tions; having  the  areas  32,  65,  and  38  square  feet,  and  the  surface 
velocities  near  the  middle  of  these  are  found  to  be  1.3,  2.6,  and 
1.4  per  second.  What  is  the  approximate  mean  velocity  of  the 
stream  ? 

ARTICLE  114.  GAUGING  BY  SUB-SURFACE  VELOCITIES. 

By  means  of  a  sub-surface  float,  or  by  a  current  meter,  the 
velocity  V  at  mid-depth  in  any  vertical  may  be  measured. 
The  mean  velocity  vl  in  that  vertical  is  very  closely 

ZY=  0.98  Vr. 

In  this  manner  the  mean  velocities  in  several  verticals  across 
the  stream  may  be  determined  by  a  single  observation  at  each 
point,  and  these  may  be  used,  as  in  Art.  112,  in  connection 
with  the  corresponding  areas  to  compute  the  discharge. 

It  was  shown  by  the  observations  of  HUMPHREYS  and  ABBOT 
on  the  Mississippi  that  the  velocity  V  is  practically  unaffected 
by  wind,  the  vertical  velocity  curves  for  different  intensities  of 
wind  intersecting  each  other  at  mid-depth.  The  mid-depth 
velocity  is  therefore  a  reliable  quantity  to  determine  and  use, 


ART.  114.]     GAUGING  BY  SUB-SURFACE    VELOCITIES.  265 

particularly  as  the  corresponding  mean  velocity  vl  for  the 
vertical  rarely  varies  more  than  I  or  2  per  cent  from  the 
value  0.98  V. 

The  following  relations  between  velocities  in  the  cross- 
section  were  also  deduced  by  HUMPHREYS  and  ABBOT.*  The 
curve  of  velocities  in  any  vertical  was  found  to  be  a  parabola 
whose  mean  equation  is 


V  =  3.26  -  0.7922 

in  which  V  is  the  velocity  at  any  distance  y  above  or  below  the 
horizontal  axis  of  the  parabola,  and  d  is  the  depth  of  the  water 
at  the  point  considered  ;  the  axis  being  at  the  distance  0.297^ 
below  the  surface.  The  depth  of  the  axis  was  found  to  vary 
greatly  with  the  wind,  an  up-stream  wind  of  force  4  depressing 
it  to  mid-depth,  and  a  down-stream  wind  of  force  5.3  elevating 
it  to  the  surface.  The  velocity  Fat  any  depth  d'  was  shown 
to  be  related  to  the  maximum  velocity  Vm  in  that  vertical  by 
the  equation 


in  which  v  is  the  mean  velocity  for  the  entire  cross-section,  and 

b=  _J^2_ 
Vd+i.$ 

These  relations  and  many  others  which  were  deduced  are  very 
interesting,  but  are  of  little  value  in  the  actual  gauging  of 
streams. 

Prob.  140.  Show  that  the  vertical  velocity  formula  of  HUM- 
PHREYS and  ABBOT  can  be  put  in  the  form 


v  =  3.19+ 0.471  ^-0.792  y, 

in  which  x  is  the  depth  below  the  surface. 

*  Physics  and  Hydraulics  of  the  Mississippi  River,  2d  Edition,  1876. 


266  FLOW  IN  RIVERS.  [CHAP.  IX. 

ARTICLE  115.  COMPARISON  OF  METHODS. 

This  chapter,  together  with  those  preceding,  furnishes  many 
methods  by  which  the  quantity  of  water  flowing  through  an 
orifice,  pipe  or  channel,  may  be  determined.  A  few  remarks 
may  now  properly  be  made  by  way  of  summary. 

The  method  of  direct  measurement  in  a  tank  is  always 
the  most  accurate,  but  except  for  small  quantities  is  expensive, 
and  for  large  quantities  is  impracticable.  Next  in  reliability 
and  convenience  come  the  methods  of  gauging  by  orifices  and 
weirs.  An  orifice  one  foot  square  under  a  head  of  25  feet  will 
discharge  about  40  cubic  feet  per  second,  which  is  as  large  a 
quantity  as  can  be  usually  profitably  passed  through  a  single 
opening.  A  weir  20  feet  long  with  a  depth  of  2.0  feet  will 
discharge  about  200  cubic  feet  per  second,  which  may  be  taken 
as  the  maximum  quantity  that  can  be  conveniently  thus 
gauged.  The  number  of  weirs  may  be  indeed  multiplied  for 
larger  discharges,  but  this  is  usually  forbidden  by  the  expense 
of  construction.  Hence  for  larger  quantities  of  water  indirect 
methods  of  measurement  must  be  adopted. 

The  formulas  deduced  for  the  flow  in  pipes  and  channels 
in  Chaps.  VII  and  VIII  enable  an  approximate  estimation  of 
their  discharge  to  be  determined  when  the  coefficients  and 
data  which  they  contain  can  be  closely  determined.  The  re- 
marks in  Art.  106  indicate  the  difficulty  of  ascertaining  these 
data  for  streams,  and  show  that  the  value  of  the  formulas  lies 
in  their  use  in  cases  of  investigation  and  design  rather  than  for 
precise  gaugings.  For  small  pipes  an  accurately  rated  water 
meter  is  a  cheap  and  convenient  method  of  measuring  the  dis- 
charge, while  for  large  pipes  it  will  often  be  found  difficult  to 
devise  an  accurate  and  economical  plan  for  precise  determina- 
tions, unless  the  conditions  are  such  that  the  discharge  may  be 
made  to  pass  over  a  weir  or  be  retained  in  a  large  reservoir 


ART.  115.]  COMPARISON  OF  METHODS.  267 

whose  capacity  is  known  for  every  tenth  of  a  foot  in  depth. 
For  large  aqueducts  and  for  canals  and  streams  the  only 
available  methods  are  those  explained  in  this  chapter. 

Surface  floats  are  not  to  be  recommended  except  for  rude 
determinations,  because  they  are  affected  by  wind,  and  because 
the  deduction  of  mean  velocities  from  them  is  always  subject 
to  much  uncertainty.  Nevertheless  many  cases  arise  in  prac- 
tice where  the  results  found  by  the  use  of  surface  floats  are 
sufficiently  precise  to  give  valuable  information  concerning 
the  flow  of  streams. 

The  double  float  for  sub-surface  velocities  is  used  in  deep 
and  rapid  rivers,  where  a  current  meter  cannot  be  well  operated 
on  account  of  the  difficulty  of  anchoring  a  boat.  In  addition  to 
its  disadvantages  already  mentioned  may  be  noted  that  of  ex- 
pense, which  becomes  large  when  many  observations  are  to 
be  taken. 

The  method  of  determining  the  mean  velocities  in  vertical 
planes  by  rod  floats  is  very  convenient  in  canals  and  channels 
which  are  not  too  deep  or  too  shallow.  The  precision  of  a 
velocity  determination  by  a  rod  float  is  always  much  greater 
than  that  of  one  taken  by  the  double  float,  so  that  the  former 
is  to  be  preferred  when  circumstances  will  allow. 

Current-meter  observations  are  those  which  now  take  the 
highest  rank  for  precision  and  rapidity  of  execution.  The  first 
cost  of  the  outfit  is  greater  than  that  required  for  rod  floats, 
but  if  much  work  is  to  be  done  it  will  prove  the  cheapest. 
The  main  objection  is  to  the  errors  which  may  be  introduced 
from  the  lack  of  proper  rating :  this  is  required  to  be  done  at 
regular  intervals,  as  it  is  found  that  the  relation  between  the 
velocity  and  the  recorded  number  of  revolutions  sometimes 
changes  during  use. 


268  FLO  IV  IN  RIVERS.  [CHAP.  IX. 

In  the  execution  of  hydraulic  operations  which  involve  the 
measurement  of  water  a  method  is  to  be  selected  which  will 
give  the  highest  degree  of  precision  with  a  given  expenditure,. 
or  which  will  secure  a  given  degree  of  precision  at  a  minimum 
expense.  Any  one  can  build  a  road,  or  a  water  supply-system  ; 
but  the  art  of  engineering  teaches  how  to  build  it  well,  and  at 
the  least  cost  of  construction  and  maintenance.  So  the  science 
of  hydraulics  teaches  the  laws  of  flow  and  records  the  results 
of  experiments,  so  that  when  the  discharge  of  a  conduit  is  to 
be  measured  or  a  stream  is  to  be  gauged  the  engineer  may 
select  that  method  which  will  furnish  the  required  information 
in  the  most  satisfactory  manner  and  at  the  least  expense. 

Prob.  141.  Devise  a  method  for  measuring  the  velocity  of 
a  current  different  from  any  described  in  the  preceding  pages* 

ARTICLE  116.  VARIATIONS  IN  VELOCITY  AND  DISCHARGE. 

When  the  stage  of  water  rises  and  falls  a  corresponding  in- 
crease or  decrease  occurs  in  the  velocity  and  discharge.  The 
relation  of  these  variations  to  the  change  in  depth  may  be 
approximately  ascertained  in  the  following  manner,  the  slope 
of  the  water  surface  being  regarded  as  remaining  uniform  : 
Let  the  stream  be  wide,  so  that  its  hydraulic  radius  is  nearly 
equal  to  the  mean  depth  d\  then 

v  =  c  Vds  = 


Differentiating  this  with  respect  to  v  and  d  gives 

6v  =  $cstd'*6d  =  %v^9 
a 

or 

dv  _      i    3d 

~~v          2    d~  ' 

Here  the  first  member  is  the  relative  change  in  velocity  when 
the  depth  varies  from  d  to  d  ±  6d,  and  the  equation  hence 


ART.  n6.]    VARIATIONS  IN    VELOCITY  AND  DISCHARGE.       269 

shows  that  the  relative  change  in  velocity  is  one-half  the  rela- 
tive change  in  depth.  For  example,  a  stream  3  feet  deep,  and 
with  a  mean  velocity  of  2  feet'  per  second,  rises  so  that  the 
depth  is  3.3  feet  ;  then 

dv=2X^X^  =  o.i, 

and  the  velocity  becomes  2  -f-  o.i  —  2.1  feet  per  second.  This 
conclusion  is  of  course  the  more  accurate  the  smaller  the  varia- 
tion Sd. 

In  the   same   manner  the  variation  in  discharge  may  be 
found.     Thus  :  let  b  be  the  breadth  of  the  stream,  then 

q  =  cbd  V~ds  = 


8q_  =  3  WB 
q         2    d  ' 

Hence  the  relative  change  in  discharge  is  ij  times  that  of  the 
relative  change  in  depth.  This  rule,  like  the  preceding,  sup- 
poses that  6d  is  very  small,  and  will  not  apply  to  large  varia- 
tions. 

The  above  conclusions  may  be  expressed  as  follows  :  If  the 
mean  depth  changes  I  per  cent,  the  velocity  changes  0.5  per 
cent,  and  the  discharge  changes  1.5  percent.  They  are  only 
true  for  streams  with  such  cross-sections  that  the  hydraulic 
radius  may  be  regarded  as  proportional  to  the  depth,  and  even 
for  such  sections  are  only  exact  for  small  variations  in  d  and  v. 
They  also  assume  that  the  slope  s  remains  the  same  after  the 
rise  or  fall  as  before  ;  this  will  be  the  case  if  a  condition  of 
permanency  is  established,  but,  as  a  rule,  while  the  stage  of 
water  is  rising  the  slope  is  increasing,  and  while  falling  it  is 
decreasing. 


2/0  FLOW  IN  RIVERS.  [CHAP.  IX. 

Prob.  142.  A  stream  of  4  feet  mean  depth  delivers  800 
cubic  feet  per  second.  What  will  be  the  discharge  when  the 
depth  is  decreased  to  3.9  feet? 


ARTICLE  117.  NON-UNIFORM  FLOW. 

In  all  the  cases  thus  far  considered,  the  slope  of  the  channel,, 
its  cross-section,  and  the  depth  of  the  water  have  been  regard- 
ed as  constant.  If  these  are  variable  along  different  reaches 
of  the  channel  the  case  is  one  of  non-uniformity,  and  the  pre- 
ceding discussions  do  not  apply  except  to  the  single  reaches. 
The  flow  being  permanent,  the  same  quantity  of  water  passes, 
each  section  per  second,  but  its  velocity  and  depth  vary  as. 
the  slope  and  cross-section  change.  To  discuss  this  case  let 
there  be  several  lengths,  £  ,  /3 ,  .  .  .  ,  ln ,  which  have  the  falls 
h^ ,  h^  ,...,//„,  the  water  sections  being  al ,  a^ ,  .  .  .  ,  an , 
the  wetted  perimeter  A ,  A  >  •  •  •  >  A>  an<^  tne  velocities  viy  v^^ 
.  .  .  ,  vn .  The  total  fall  hl  -f-  h^  -\-  .  .  .  +  ^«  is  expressed  by 
h.  Now  the  head  corresponding  to  the  mean  velocity  in  the 

first  section  is  — .     The  theoretic  head  for  the  last  section  is 
2£ 

v*  v  2 

h-\ — L,  while  the  actual  velocity-head  is  — .     The  difference 

between  these  is  the  head  lost  in  friction,  or 

k    vL_^-.fhL^-A  fM^<         fM^ 

^2g       2g  -7'  a,    2g  +7°  a.   2g  ^  t"7"  a.  2g  ' 

in  which  /, ,  f9 ,  .  .  .  ,  fn  are  the  friction  factors  for  the  differ- 
ent sections  and  surfaces,  whose  values  in  terms  of  the  velocity 
coefficient  c  are,  as  seen  from  Art.  94, 

f    _  2S        f  _  2<T  f  __  *g 

J  \   —       a>        ya  —       «>       •••»      Jn  —  ~~T  • 


ART.  H7.]  NON-UNIFORM  FLOW.  2/1 

Let  q  be  the  discharge  per  second  ;  then,  as  the  flow  is  perma- 
nent, 


Inserting  in  the  equation  these  values  of  /and  v,  it  becomes 


which  is  a  fundamental  formula  for  the  discussion  of  the  flow 
in  non-uniform  channels.  Since  the  values  of  c  given  in  this 
chapter  are  for  English  feet,  the  data  of  numerical  problems 
can  be  inserted  only  when  expressed  in  the  same  unit. 

The  above  discussion  shows  that  the  discharge  q  is  a  con- 
sequence, not  only  of  the  total  fall  h  in  the  entire  length  of 
the  channel,  but  also  of  the  dimensions  of  the  various  cross- 
sections.  The  assumption  has  been  made  that  a  and  /  are 
constant  in  each  of  the  parts  considered  ;  this  can  be  realized 
by  taking  the  lengths  /!,/„,  etc.,  sufficiently  short.  If  only 
one  part  be  considered  in  which  a  and  p  are  constant,  an  —  al  , 
all  the  terms  but  one  in  the  second  member  disappear,  and  the 
two  equations  reduce  to  the  simple  formulas  previously  de- 
duced for  the  velocity  and  discharge  in  a  uniform  channel. 

An  interesting  problem  is  that  where  the  flow  is  non- 
uniform  in  a  channel  of  constant  slope  and  section,  which  may 
be  caused  by  an  obstruction  in  the  stream  above  or  below  the 
part  considered.  Here  let  al  and  a^  be  two  sections  whose 
distance  apart  is  /,  and  let  vl  and  v^  be  the  mean  velocities  in 
those  sections.  Then  if  a  and  /  be  average  values  of  the 
wetted  area  and  perimeter,  the  formula  becomes 


FLOW  IN  RIVERS. 


[CHAP.  IX. 


from  which  q  can  be  computed  when  the  other  quantities  are 

known.  The  important  problem, 
however,  is  to  discuss  the  change 
in  depth  between  the  two  sec- 
tions. For  this  purpose  let 
A^A^  in  Fig.  76  be  the  longi- 
tudinal profile  of  the  water  sur- 


FIG.  76. 

face,  let  A,D  be  horizontal,  and  A,C  be  drawn  parallel  to  the 
bed  B^^.  The  depths  A^B^  and  A^  are  represented  by  dl 
and  4 ,  the  latter  being  taken  as  the  larger.  Let  i  be  the  con- 
stant slope  of  the  bed  B& ;  then  DC  =  z'/,  and  since  DA^  =±  h 
and  Af  =  4  —  dl , 

h  =  il-  (4-4). 

Inserting  this  value  of  h  in  the  above  equation  and  solving 'for 
/,  there  results 


q"P 


.    .    (76) 


from  which  the  length  /  corresponding  to  a  change  in  depth 
4  —  4  can  ke  approximately  computed.  This  formula  is  the 
more  accurate  the  shorter  the  length  /,  since  then  the  mean 
quantities/  and  a  can  be  obtained  with  greater  precision,  and 
c  is  subject  to  less-variation. 

The  inverse  problem,  to  find  the  change  in  depth  when  /  is 
given,  cannot  be  directly  solved  by  this  formula,  because  the 
areas  are  functions  of  the  depths.  If  the  change  is  not  great, 
however,  a  solution  may  be  effected  for  the  case  of  a  channel 
whose  breadth  b  is  constant  by  regarding/  and  a  as  equal  to/t 
and  tfj  ;  and  also  by  putting 

I  I         a? -a?       4" -4' 


ART.  nS.]  THE   SURFACE   CURVE.  2/3 

The  formula  then  becomes 

•          o  p 

If      ^^      ~r      n T" 


I     — 


from  which  d^  can  be  approximately  computed  when  all  the 
other  quantities  are  given. 

Fig.  76  is  drawn  for  the  case  of  depth  increasing  down 
stream,  but  the  reasoning  is  general,  and  the  formulas  apply 
equally  well  when  the  depth  decreases.  In  the  latter  case  the 
point  Az  is  below  C,  and  dz  —  dl  will  be  found  to  be  nega- 
tive. As  an  example,  let  it  be  required  to  determine  the  de- 
crease in  depth  in  a  rectangular  conduit  5  feet  wide  and  333 
feet  long,  which  is  laid  with  its  bottom  level,  the  depth  of  water 
at  the  entrance  being  maintained  at  2  feet,  and  the  quan- 
tity supplied  being  20  cubic  feet  per  second.  Here  /  =  333, 
^  —  5>  di  =  2>  Pi  —  5  +  4  =  9>  #  =  20,  and  i  =  o.  Taking 
c  =  89,  and  substituting  all  values  in  the  formula,  there  is 
found  d^  —  d^  —  —  0.16  feet,  whence  d^  —  1.84  feet,  which  is 
to  be  regarded  as  an  approximate  probable  value.  It  is  likely 
that  values  of  d^  —  dl  computed  in  this  manner  are  liable  to  an 
uncertainty  of  10  or  20  per  cent,  the  longer  the  distance  /  the 
greater  being  the  error  of  the  formula.  In  strictness  also  c 
varies  with  depth,  but  errors  from  this  cause  are  small  when 
compared  to  those  arising  in  selecting  its  value. 

Prob.  143.  Compute  the  value  of  d^  for  the  above  example 
when  the  bed  of  the  conduit  has  the  uniform  slope  i  =  o.oi. 

Ans.  4  =  5.38  feet. 

ARTICLE  118.  THE  SURFACE  CURVE. 

In  the  case  of  uniform  flow  the  slope  of  the  water  surface  is 
parallel  to  that  of  the  bed  of  the  channel,  and  the  longitudinal 


274 


FLO  IV  IN 


[CHAP.  IX. 


profile  of  the  water  surface  is  a  straight  line.  In  non-uniform 
flow,  however,  the  slope  of  the  water  surface  continually  varies, 
and  the  longitudinal  profile  is  a  curve  whose  nature  is  now  to 
be  investigated.  As  in  the  last  article,  the  slope  i  of  the  bed 
of  the  channel  will  be  taken  as  constant,  and  its  cross-section 
will  be  regarded  as  rectangular.  Moreover,  it  will  be  assumed 
that  the  stream  is  wide  compared  to  its  depth,  so  that  the 
wetted  perimeter  may  be  taken  as  equal  to  the  width  and  the 
hydraulic  radius  equal  to  the  mean  depth  (Art.  93).  These 
assumptions  are  closely  fulfilled  in  many  canals  and  rivers. 

The  last  formula  of  the  preceding  article  is  rigidly  exact  if 
the  sections  al  and  #a  are  consecutive,  so  that  /  becomes  <5/ 
and  d^  —  d^  becomes  dd.  Making  these  changes,  and  placing 

-  equal  to  -j,  in  accordance  with  the  above  assumptions,  the 
formula  becomes 


(77) 


in  which  d  is  the  depth  of  the  water  at  the  place  considered. 
This  is  the  general  differential  equation  of  the  surface  curve. 

To  discuss  this  curve  let  D  be  the  depth  of  the  water  if  the 

flow  were  uniform.  The  slope 
s  of  the  water  surface  would 
then  be  equal  to  the  slope  i  of 
the  bed  of  the  channel,  and 
from  the  general  equation  for 
mean  velocity, 

q  —  av  —  cbD  VTt  =  cbD  VIM. 
Inserting   this  value"  of  q  the 
FIG.  77.  equation  reduces  to 


ART.  iiS.]  THE  SURFACE   CURVE.  2?$ 


: 


in  which  d  and  /  are  the  only  variables,  the  former  being  the 
ordinate  and  the  latter  the  abscissa,  measured  parallel  to  the 
bed  BB,  of  any  point  of  the  surface  curve. 

First,  suppose  that  D  is  less  than  d,  as  in  the  upper  diagram 
of  Fig.  77,  where  AA  is  the  surface  curve  under  the  non-uni- 
form flow,  and  CC  is  the  line  which  the  surface  would  take  in 
case  of  uniform  flow.  The  numerator  of  (77)'  is  then  positive, 
and  the  denominator  is  also  positive,  since  i  is  very  small. 
Hence  6d  is  positive,  and  it  increases  with  d  in  the  direction 
of  the  flow ;  going  up  stream  it  decreases  with  d^ ,  and  the  sur- 
face curve  becomes  tangent  to  CC  when  d  =  D.  This  form  of 
curve  is  that  usually  produced  above  a  dam,  and  is  called  the 
curve  of  backwater. 

Second,  let  d  be  less  than  D,  as  in  the  second  diagram  of 
Fig.  74.  The  numerator  is  then  negative,  and  the  denomi- 
nator positive  ;  6d  is  accordingly  negative,  and  A  A  is  concave 
to  the  bed  BB,  whereas  in  the  former  case  it  was  convex. 
This  form  of  surface  curve  may  occur  when  a  sudden  fall  exists 
in  the  stream  below  the  point  considered ;  it  is  of  slight  practi- 
cal importance  compared  to  the  previous  case. 

A  very  curious  phenomenon  is  that  of  the  so-called  "jump" 
which  sometimes  occurs  in  shallow  channels,  as  shown  in  Fig. 
78.  This  happens  when  the  de- 
nominator in  (77)'  is  zero,  the  ^^^—i 
numerator  being  positive ;  then 

p>    r 

-j-r   becomes     infinite,    and   the 

O/  FIG.  78. 


2/6  FLOW  IN  RIVERS.  [CHAP.  IX. 

water  surface  stands  normal  to  the  bed.     Placing  the  denomi- 
nator of  (77)  equal  to  zero,  there  is  found 


or         =g. 

Now  by  further  consideration  it  will  appear  that  the  varying 
denominator  in  passing  through  zero  changes  its  sign.  Above 
the  jump  where  the  depth  is  dl  the  velocity  is  greater  than 
Vgdi  ,  and  below  it  is  less  than  Vgd±  .  The  condition  for  the 
occurrence  of  the  jump  is  that  an  obstruction  should  exist  in 
the  stream  below,  that  the  slope  i  should  not  be  small,  and 
that  the  velocity  should  be  greater  than  Vgdl  .  To  find  the 
slope  i  which  is  necessary, 

vl=cVdli       v*>gd^       whence   i  >—  . 

Hence  the  jump  cannot  occur  when  i  is  less  than  C,    For  an 

unplaned  plank  trough  c  may  be  taken  at  about  100  ;  hence  the 
slope  for  this  must  be  equal  to  or  greater  than  0.00322. 

To  determine  the  height  of  the  jump,  or  the  value  of  d^  in 


v*       v* 


terms  of  d^ ,  it  is  to  be  observed  that  the  lost  head  is  — — , 

and  that  this  is  lost  in  two  ways,  first  by  the  impact  due  to  the 
enlargement  of  section  (Art.  68),  and  second  by  the  rising  of  the 
whole  quantity  of  water  through  the  height  J(^2  —  d^),  the  loss 
in  friction  in  the  short  distance  between  dl  and  </3  being  neg- 
lected. Hence 

2  2  /  \'2  J  J 

Vi  —  v*  __  (Vi  —  v*)    ,a*  —  ai 

2g  2g  2 

Inserting  in  this  the  value  of  z/a,  found  from  the  relation 
v^  =  z/j4  >  dividing  by  d^  —  dl ,  and  solving  for  d^ ,  gives 

(78) 


Vi. 

Observed  d*  . 

Computed  d*  . 

4.59 

0.423 

0439 

447 

5-59 
6.28 

0.421 
0.613 
0-739 

0437 
0.636 
0.777 

ART.  iig.]  BACKWATER.  277 

The  following  is  a  comparison  between  the  values  of  d^  com- 
puted by  this  formula  and  the  observed  values  in  four  experi- 
ments made  by  BlDONE,  the  depths  being  in  feet : 


0.149 
0.154 
0.208 
0.246 


The  agreement  is  very  fair,  the  computed  values  being  all 
slightly  greater  than  the  observed,  which  should  be  the  case, 
because  the  above  reasoning  omits  the  frictional  resistances 
between  the  points  where  dl  and  d^  are  measured. 

Prob.  144    Discuss  formula  (78)  by  placing  for  q  its  value 
cbd  V~ds,  where  s  is  the  slope  of  the  water  surface. 


ARTICLE  119.  BACKWATER. 

When  a  dam  is  built  across  a  channel  the  water  surface  is 
raised  for  a  long  distance  up  stream.  This  is  a  fruitful  source 
of  contention,  and  accordingly  many  attempts  have  been  made 
to  discuss-  it  theoretically,  in  order  to  be  able  to  compute  the 
probable  increase  in  depth  at  various  distances  back  from  a 
proposed  dam.  None  of  these  can  be  said  to  have  been  suc- 
cessful except  for  the  simple  case  where  the  slope  of  the  bed 
of  the  channel  is  constant,  and  its  cross-section  such  that  the 
width  may  be  regarded  as  uniform  and  the  hydraulic  radius 
be  taken  as  equal  to  the  depth.  These  conditions  are  closely 
fulfilled  for  many  streams,  and  an  approximate  solution  may  be 
made  by  the  formula  (77)  of  Art.  118.  It  is  desirable,  how- 
ever, to  obtain  an  exact  equation  of  the  surface  curve,  so  as  to 
secure  a  more  reliable  method. 


2/8  FLOW  IN  RIVERS.  [CHAP.  IX. 

For  this  purpose  the  differential  equation  (77)'  of  the  last 
article  may  be  written  in  the  form 


in  which  /  and  d  are  the  co-ordinates  of  any  point  of  the  curve. 
Let  -=r  be  the  independent  variable  x,  so  that  d  =  Dx ;  then 

D  D(         W\     6x 

i  i\          g  I  x*  —  i  ' 

the  general  integral  of  which  is 

Dx      D(         ^i\f\  ,       x*  +  x-\-i         i  2*4 

/  =  — r[  i 1(  7-  log,  -7 rj ;=  arc  cot  — -: 

^         ^  \         g J \6  (^  —  J)          rs  V^ 

which  is  the  equation  of  the  surface  curve.     To  use  this  let 

4>(x)  or  0  Ky )  be  put  as  an  ab- 

I  , -r --p-,^:——--. 

J  1 1  p      ^^.^^  breviation  for  the  logarithmic 

v/^//'  /////////////////////////g  and    circular    function    in    the 

FlG-  79.  second  member.     Also  let  d^  be 

the  depth  at  the  dam,  and  let  /  be  measured  up  stream  from 

that  point  to  a  section  where  the  depth  is  dl .     Then  taking 

the  integral  between  these  limits  the  equation  becomes 


'= 


(7  -I)  [«(£}- 


which  is  the  practical  formula  for  use.  In  like  manner;  d^  may 
represent  a  given  depth  at  any  section,  and  dl  any  depth  farther 
up  the  stream. 

When  d  =.  D,  or  the  depth  of  the  backwater  becomes  equal 
to  that  of  the  previous  uniform  flow,  x  is  unity,  and  hence  /  is 


ART.  119.]  BACKWATER.  2/9 

infinity.  The  slope  CC  of  uniform  flow  is  therefore  an  asymp- 
tote to  the  backwater  curve.  Accordingly,  no  matter  how 
little  d^  may  exceed  D,  the  depth  dl  is  always  greater  than  D, 
although  it  often  happens  for  steep  slopes  that  dl  becomes 
practically  equal  to  D  at  distances  above  d^  ,  which  are  not 
great.  In  the  investigations  of  backwater  problems  there  are 
two  cases  :  dl  and  d^  may  be  given  and  /  is  to  be  found,  or  /  is 
given  and  one  of  the  depths  is  to  be  found.  To  solve  these 
problems,  a  table  giving  values  cf  the  backwater  function 

found  on  the  next  page.*     The  argument  of  the 


table  is  -r  ,  which  being  less  than  unity,  is  more  convenient  for 

tabular  purposes  than  -=:  ,  whose  values  range  from  o  to    oo. 
The  following  examples  will  illustrate  the  method  of  procedure. 

A  stream  of  5  feet  depth  is  to  be  dammed  so  that  the  water 
just  above  the  dam  will  be  10  feet.  Its  uniform  slope  is 
0.000189,  or  a  little  less  than  one  foot  per  mile,  and  the  surface 
of  its  channel  is  such  that  the  coefficient  c  is  65.  It  is  required 
to  find  the  distance  back  from  the  dam  at  which  the  depth  of 

water  is  6   feet.     Here  dz  =  10,  dl  =  6,  D  =  5,  —  -  =  0.5   for 
which  the  table  gives  0Uj)  =0.1318,   ~r  "=•  0.833  f°r  which 

the  table  gives  <f>  \-^\  =  0.4792,  and  —  —  5291.     These  values 
inserted  in  the  formula  give 

/  =  (10  -  6)5291  +  5  (5291  -  ^ig)  (0.4792  -  0.1318); 
/=  30  125  feet  =  5.70  miles. 

*From  BRESSE'S  La  Mecanique  Appliquee  (Paris,  1873),  vol.  ii.  p.  556. 


28O  FLO  W  IN  RIVERS.  [CHAP.  IX. 

TABLE  XXII.  VALUES  OF  THE  BACKWATER  FUNCTION. 


D 
~d 

*d) 

D 
d 

<(!) 

D 
~d 

HI) 

D 
d 

HI) 

I, 

CO 

0.948 

0.8685 

0.815 

0.4454 

0.52 

0.1435 

0.999 

2.1834 

.946 

.8539 

.810 

•4367 

•51 

.1376 

.998 

1-9532 

•944 

.8418 

.805 

.4281 

•50 

.1318 

•997 

1.8172 

.942 

.8301 

.800 

.4198 

•49 

.1262 

.996 

I.72I3 

.940 

.8188 

•795 

.4117 

•48 

.1207 

•995 

I  .  6469 

.938 

.8079 

•  790 

•4039 

•  47 

•  1154 

•994 

1.5861 

•936 

•7973 

.785 

.3962 

•46 

.IIO2 

•993 

1.5348 

•934 

.7871 

.780 

.3886 

•45 

.1052 

.992 

1.4902 

•932 

•7772 

•775 

.3813 

44 

.1003 

.991 

I.45IO 

•930 

.7675 

•770 

•3741 

•43 

•0995 

.990 

I.4I59 

.928 

•758i 

•765 

.3671 

•42 

.0909 

.989 

1.3841 

.926 

.7490 

.760 

.3603 

.41 

.0865 

.988 

I.355I 

.924 

.7401 

•755 

.3526 

.40 

.0821 

.987 

1.3248 

.922 

•7315 

•  750 

•3470 

•39 

.0779 

.986 

1.3037 

.920 

.7231 

•745 

.3406 

•38 

.0738 

•985 

1.2807 

.918 

.7149 

•740 

•3343 

•37 

.0699 

.984 

1.2592 

.916 

.7069 

•735 

.3282 

.36 

.0666 

•9S3 

1.2390 

.914 

.6990 

•730 

-3221 

•35 

.0623 

.982 

1.2199 

.912 

.6914 

.725 

.3162 

.34 

.0587 

.981 

1.2019 

.910 

•  6839 

.720 

.3104 

•33 

•0553 

.980 

1.1848 

.908 

.6766 

.715 

•3047 

•32 

.0519 

•979 

1.1686 

.906 

.6695 

.710 

.2991 

•31 

.0486 

.978 

I.I53I 

.904 

.6625 

•705 

•2937 

•30 

•0455 

•977 

I.I383 

.902 

.6556 

.70 

.2883 

.29 

.0425 

.976 

1.1241 

.900 

.6489 

.69 

.2778 

.28 

•0395 

•  975 

1.1105 

.895 

.6327 

.68 

.2677 

.27 

.0367 

•  974 

1.0974 

.890 

.6173 

.67 

.2580 

.26 

.0340 

•  973 

1.0848 

.885 

.6025 

.66 

.2486 

.25 

.0314 

•  972 

1.0727 

.880 

.5884 

•65 

•2395 

.24 

.0290 

.971 

I.  0610 

•875 

•5749 

.64 

.2306 

•  23 

.0266 

.970 

1.0497 

.870 

.5619 

•63 

.2221 

.22 

.0243 

.968 

1.0282 

.865 

•5494 

.62 

.2138 

.21 

.0221 

.966 

1.0080 

.860 

•5374 

.61 

.2058 

.20 

.O2OI 

•964 

0.9890 

.855 

.5258 

.60 

.1980 

.18 

.  .Ol62 

.962 

.9700 

.850 

•  5146 

•59 

.1905 

.16 

.0128 

.960 

•9539 

.845 

.5037 

.58 

.1832 

•14 

.0098 

•  958 

.9376 

.840 

•  4932 

•57 

.1761 

.12 

.0072 

.956 

.9221 

•835 

.4831 

•56 

.  1692 

.  IO 

.0050 

•954 

.9073 

.830 

•  4733 

•55 

•  1625 

.06 

.0018 

•952 

.8931 

.825 

•  4637 

•54 

.1560 

.01 

.0001 

•950 

•8795 

.820 

•  4544 

•53 

.1497 

.00 

.OOOO 

ART.  119.]  BACKWATER.  28l 

In  this  case  the  water  is  raised  one  foot  at  a  distance  of  5.7 
miles  up  stream  from  the  dam,  in  spite  of  the  fact  that  the  fall 
in  the  bed  of  the  channel  is  nearly  5.7  feet. 

The  inverse  problem,  to  compute  d^  or  d^ ,  when  /  and  d^  or 
dl  is  given,  can  only  be  solved  by  repeated  tentative  trials  by 
the  help  of  Table  XXII.  For  example,  let  /  =  30  125  feet,  the 
other  data  as  above,  and  it  be  required  to  determine  d^  so  that 
dl  shall  be  only  5.2  feet,  or  0.2  feet  greater  than  the  original 

D        5 

depth  of  5  feet.     Here  —  =  --  =  0.962,  and  from  the  table 

*i      5-2 

}  =  0.9700.     Then  the  formula  becomes 

30  125  =  (4  -  5-2)5291  +  5  X  5i6o[o.97oo  -  0(^ 
which  reduces  to 

32  610  =  5291^  -  25  8oo0(~)  . 
Values  of  d^  are  now  to  be  assumed  until  one  is  found  which 

satisfies  this  equation.     Let  d^  =  8  feet,  then  -7-  =  0.625,  and 

"2 

from  the  table  0(  j£j  =  0.2179.     Substituting  these, 

32  610  =  42  328  —  5  622  =  36  706, 
which  shows  that  the  assumed  value  is  too  large.     Again,  take 

d^  =  7  feet,  then  -j-  =  0.714,  and  from  the  table  0(-^)  =  0.3036. 

whence 

32  610  =  37  037  —  7  833  =  29  204, 

which  shows  that  7  feet  is  too  small.     If  d^  =  7.4  feet, 

—  =  0.675     and     0V7?]  —  0.2628, 

and  then 

32  610  =  39  153  -  6  780  =  32  373. 


282  FLOW  IN  RIVERS.  [CHAP.  IX. 

This  indicates  that  7.4  is  a  little  too  small,  and  on  trying  7.5  it 
is  found  to  be  too  large.  The  value  of  d^  hence  lies  between 
7.4  and  7.5  feet,  which  is  as  close  a  solution  as  will  generally  be 
required.  The  height  of  the  dam  may  now  be  computed  by 
Art.  58,  taking  the  rise  d'  at  about  2.45  feet. 

In  conclusion,  it  should  be  said  that  if  the  slope,  width,  or 
depth  changes  materially,  the  above  method  cannot  be  em- 
ployed in  which  the  distance  /  is  counted  from  the  dam  as  an 
origin.  In  such  cases  the  stream  should  be  divided  into  reaches, 
for  each  of  which  these  quantities  can  be  regarded  as  constant. 
The  formula  can  then  be  used  for  the  first  reach,  and  the  depth 
at  its  upper  section  determined  ;  calling  this  depth  d^ ,  the  ap- 
plication can  then  be  made  to  the  next  reach,  and  so  on  in 
order.  Strictly  speaking,  the  coefficient  c  varies  with  the 
depth,  and  by  KUTTER's  formula  (Art.  101)  its  varying  values 
may  be  ascertained,  if  it  be  thought  worth  the  while.  Even  if 
this  be  done,  the  resulting  computations  must  be  regarded  as 
liable  to  considerable  uncertainty.  In  computing  depths  for 
given  lengths  probably  an  uncertainty  of  10  per  cent  or  more 
in  values  of  d^  —  dl  should  be  expected.  In  regard  to  the 
depth  D,  it  may  be  said  that  this  should  be  determined  by  the 
actual  measurement  of  the  area  and  wetted  perimeter  of  the 
cross-section  during  uniform  flow,  the  hydraulic  radius  com- 
puted from  these  being  taken  as  D. 

Prob.  145.  A  stream  whose  cross-section  is  2400  square 
feet  and  wetted  perimeter  300  feet  has  a  uniform  slope  of  2.07 
feet  per  mile,  and  its  condition  is  such  that  c  =  70.  It  is  pro- 
posed to  build  a  dam  which  raises  the  water  6  feet  above  its 
former  level,  without  increasing  its  width.  Compute  the 
amount  of  rise  due  to  the  backwater  at  distances  of  I,  2,  and  3 
miles  up  stream  from  the  dam. 


ART.  120.]        THEORETIC  AND  EFFECTIVE  POWER.  283 


CHAPTER  X. 
MEASUREMENT  OF  WATER  POWER. 

ARTICLE  120.  THEORETIC  AND  EFFECTIVE  POWER. 

The  theoretic  energy  of  W  pounds  of  water  falling  through 
h  feet  is  Wh  foot-pounds,  and  if  this  occurs  in  one  second  the 
energy  per  second  is  Wh,  and  the  theoretic  horse-power  is 


HP=         =  0.001818  Wh  .....     (80) 


If  this  power  could  all  be  utilized  it  would  be  able  to  lift  the 
same  weight  of  water  per  second  through  the  same  vertical 
height,  and  an  efficiency  of  unity  would  be  secured.  Owing 
to  friction,  impact,  leakage,  and  other  losses,  the  efficiency 
must  always  be  less  than  unity. 

When  tjie  energy  of  a  water-fall  is  to  be  transformed  into 
useful  work  the  water  is  made  to  pass  over  a  wheel  or  through 
a  hydraulic  motor  in  such  a  manner  that  when  the  fall  h  has 
been  accomplished  the  water  has  little  or  no  velocity.  If  the 
water  falls  freely  through  the  height  h  it  acquires  the  velocity 
V2gh,  and  the  energy  is  still  potential,  and  equal  to  Wh.  If, 
however,  this  velocity  is  destroyed,  the  energy  is  either  trans- 
formed into  heat  or  into  useful  work.  In  the  case  of  flow 
through  a  long  pipe  nearly  all  the  energy  of  the  head  h  may 
be  expended  in  heat  in  overcoming  the  frictional  resistances; 
such  a  method  of  bringing  water  to  a  motor  is  therefore  to 
be  avoided. 

To  utilize  the  energy  of  a  water-fall  in  work  the  water  is  to 
be  collected  in  a  reservoir,  canal,  or  head  race,  from  which  it  is 


284  MEASUREMENT  OF    WATER  POWER.  [CHAP.  X. 

carried  to  the  motor  through  a  pipe,  penstock,  or  flume,  and 
after  doing  its  work  it  issues  into  the  tail  race  or  lower  level. 
In  designing  these  constructions  care  should  be  taken  to  avoid 
losses  in  energy  or  head,  and  for  this  purpose  the  principles  of 
the  preceding  chapters  should  be  applied.  The  entrance  from 
the  head  race  into  the  penstock,  and  from  the  penstock  to  the 
motor,  should  be  smooth  and  well  rounded ;  sudden  changes  in 
cross-section  should  be  avoided,  and  all  velocities  should  be  low 
except  that  which  is  to  be  utilized  in  impulse.  If  these  pre- 
cautions be  carefully  observed  the  loss  in  the  head  h  outside  of 
the  motor  can  be  made  very  small. 

The  effective  power  of  a  water-fall,  or  that  utilized  by  the 
motor,  may  be  in  the  best  constructions  as  high  as  90  per  cent 
of  the  theoretic  power.  In  any  case,  if  e  be  the  efficiency  and 
k  the  work  actually  obtained, 

k^eK  —  eWh. 

That  hydraulic  motor  will  be  the  best,  other  things  being  equal, 
which  furnishes  the  highest  value  of  e.  In  practice  values  of  e 
are  usually  between  the  limits  0.25  and  0.90,  the  lower  values 
occurring  where  a  cheap  and  abundant  water  supply  exists,  so 
that  sufficient  power  can  be  obtained  with  an  inexpensive 
wheel,  for  it  is  a  general  rule  that  the  cost  of  a  hydraulic  motor 
increases  with  the  efficiency. 

There  are  to  be  distinguished  two  efficiencies — the  effici- 
ency of  the  fall  and  the  efficiency  of  the  motor.  The  same 

expression 

k 

=  W/t 

will  apply  to  both,  k  being  the  effective  work  of  the  motor. 
The  efficiency  of  the  fall  is  that  value  of  e  found  by  using  the 
actual  weight  W  delivered  per  second,  and  the  total  height  h 
from  the  water  level  in  the  head  race  to  that  in  the  tail  race. 


ART.  i2i.]  MEASURRMENT  OF   THE    WATER.  285 

The  efficiency  of  the  motor  is  that  value  of  e  found  by  using 
the  actual  weight  W  which  passes  through  the  motor,  and  the 
effective  head  h  that  acts  upon  it.  A  The  second  W  may  be  less 
than  the  first  on  account  of  leakage,  and  the  second  Ji  may  be 
less  than  the  first  on  account  of  losses  of  head. 

To  determine  the  theoretic  power  in  any  case,  it  is  only 
necessary  to  measure  W  and  h  and  insert  their  values  in  for- 
mula (80).  The  two  following  articles  will  treat  of  these  meas- 
urements, and  the  determination  of  the  effective  power  of  the 
motor  will  be  discussed  afterwards.  The  efficiency  e  is,  then, 
the  ratio  of  the  effective  power  to  the  theoretic  power,  or  the 
ratio  of  the  effective  work  to  the  theoretic  work. 

Prob.  146.  A  weir  with  end  contractions  and  no  velocity 
of  approach  has  a  length  of  1.33  feet,  and  the  depth  on  the 
crest  is  0.406  feet.  The  same  water  passes  through  a  small 
turbine  under  the  effective  head  10.49  ^eet-  Compute  the 
theoretic  horse-power.  Ans.  1.28. 

ARTICLE  121.  MEASUREMENT  OF  THE  WATER. 

In  order  to  determine  the  weight  W  which  is  delivered  per 
second  there  must  be  known  the  discharge  per  second  q,  and 
the  weight  of  a  cubic  unit  of  water,  or 

W  =  wq. 

The  quantity  w  is  to  be  found  by  weighing  very  accurately  one 
cubic  foot,  or  any  given  volume  of  water,  or  by  noting  the  tem- 
perature and  using  the  table  in  Art.  3.  In  common  approxi- 
mate computations  w  may  be  taken  at  62.5  pounds  per  cubic 
foot.  In  precise  tests  of  motors,  however,  its  actual  value 
should  be  ascertained  as  closely  as  possible. 

The  measurement  of  the  flow  of  water  through  orifices, 
weirs,  tubes,  pipes,  and  channels  has  been  so  fully  discussed  in 


286  MEASUREMENT   OF    WATER  POWER.  [CHAP.  X. 

the  preceding  chapters,  that  it  only  remains  here  to  mention 
one  or  two  simple  methods  applicable  to  small  quantities,  and 
to  make  a  few  remarks  regarding  the  subject  of  leakage.  In 
any  particular  case  that  method  of  determining  q  is  to  be  se- 
lected which  will  furnish  the  required  degree  of  precision  with 
the  least  expense  (Art.  1 1 5). 

For  a  small  discharge  the  water  may  be  allowed  to  fall  into 
a  tank  of  known  capacity.  The  tank  should  be  of  uniform 
horizontal  cross-section,  whose  area  can  be  accurately  deter- 
mined, and  then  the  heights  alone  need  be  observed  in  order 
to  find  the  volume.  These  in  precise  work  will  be  read  by 
hook  gauges,  and  in  cases  of  less  accuracy  by  measurements 
with  a  graduated  rod.  At  the  beginning  of  the  experiment  a 
sufficient  quantity  of  water  must  be  in  the  tank  so  that  a  read- 
ing of  the  gauge  can  be  taken  ;  the  water  is  then  allowed  to 
flow  in,  the  time  between  the  beginning  and  end  of  the  experi- 
ment being  determined  by  a  stop-watch,  duly  tested  and  rated. 
This  time  must  not  be  short,  in  order  that  the  slight  errors  in 
reading  the  watch  may  not  affect  the  result.  The  gauge  is 
read  at  the  close  of  the  test  after  the  surface  of  the  water  be- 
comes quiet,  and  the  difference  of  the  gauge-readings  gives  the 
depth  which  has  flowed  in  during  the  observed  time.  The 
depth  multiplied  by  the  area  of  the  cross-section  gives  the  vol- 
ume, and  this  divided  by  the  number  of  seconds  during  which 
the  flow  occurred  furnishes  the  discharge  per  second  q. 

If  the  discharge  be  very  small,  it  may  be  advisable  to  weigh 
the  water  rather  than  to  measure  the  depths  and  cross-sections. 
The  total  weight  divided  by  the  time  of  flow  then  gives  directly 
the  weight  W.  This  has  the  advantage  of  requiring  no  tem- 
perature observation,  and  is  probably  the  most  accurate  of  all 
methods,  but  unfortunately  it  is  not  possible  to  weigh  a  con- 
siderable volume  of  water  except  at  great  expense. 


ART.  I2i.]  MEASUREMENT  OF   THE    WATER.  287 

When  water  is  furnished  to  a  motor  through  a  small  pipe 
a  water  meter  may  often  be  advantageously  used  to  determine 
the  discharge.  This  consists  of  a  box  with  two  chambers,  the 
water  entering  into  one  and  passing  out  of  the  other.  In  going 
from  the  first  to  the  second  chamber  the  water  moves  a  vane, 
a  piston,  a  disk,  or  some  other  device,  which  communicates 
motion  to  a  train  of  clockwork,  and  thereby  causes  pointers 
to  move  on  dials.  The  external  appearance  of  a  water  meter  is 
similar  to  that  of  a  gas  meter,  and  it  is  read  in  the  same  way. 
No  water  meter,  however,  can  be  regarded  as  accurate  until  it 
has  been  tested  by  comparing  the  discharge  as  recorded  by  it 
with  the  actual  discharge  as  determined  by  measurement  or 
weighing  in  a  tank.  Such  a  test  furnishes  the  constants  for 
correcting  the  result  found  by  its  readings,  which  otherwise  is 
liable  to  be  5  or  10  per  cent  in  error. 

The  leakage  which  occurs  in  the  flume  or  penstock  before 
the  water  reaches  the  wheel  should  not  be  included  in  the 
value  of  W,  which  is  used  in  computing  its  efficiency.  The 
manner  of  determining  the  amount  of  leakage  will  vary  with 
the  particular  circumstances  of  the  case  in  hand.  If  it  be  very 
small,  it  may  be  caught  in  pails  and  directly  weighed.  If  large 
in  quantity,  the  gates  which  admit  water  to  the  wheel  may  be 
closed,  and  the  leakage  being  then  led  into  the  tail  race  it  may 
be  there  measured  by  a  weir,  or  by  allowing  it  to  collect  in  a 
tank.  The  leakage  from  a  vertical  penstock  whose  cross-sec- 
tion is  known  may  be  ascertained  by  filling  it  with  water,  the 
wheel  being  still,  and  then  observing  the  fall  of  the  \vater  level 
at  regular  intervals  of  time.  In  designing  constructions  to 
bring  water  to  a  motor,  it  is  best,  of  course,  to  arrange  them  so 
that  all  leakage  will  be  avoided,  but]  this  cannot  often  be  fully 

attained,  except  at  great  expense. 

0 

The  most  common  method  of  measuring  q  is  by  means  of 
a  weir  placed  in  the  tail  race  below  the  wheel.  This  has  the 


288  MEASUREMENT  OF    WATER  POWER.          [CHAP.  X. 

disadvantage  that  it  sometimes  lessens  the  fall  which  would  be 
otherwise  available,  and  that  often  the  velocity  of  approach  is 
high.  It  has,  however,  the  advantage  of  cheapness  in  construc- 
tion and  operation,  and  for  any  considerable  discharge  appears 
to  be  almost  the  only  method  which  is  both  economical  and 
precise. 

Prob.  147.  A  vertical  penstock  whose  cross-section  is  15.98 
square  feet  is  filled  with  water  to  a  depth  of  10.50  feet.  Dur- 
ing the  space  of  two  minutes  the  water  level  sinks  0.02  feet. 
What  is  the  leakage  in  cubic  feet  per  second  ? 

ARTICLE  122.  MEASUREMENT  OF  THE  HEAD. 

The  total  available  head  h  between  the  surface  of  the  water 
in  the  reservoir  or  head  race  and  that  in  the  lower  pool  or  tail 
race  is  determined  by  running  a  line  of  levels  from  one  to  the 
other.  Permanent  bench  marks  being  established,  gauges  can 
then  be  set  in  the  head  and  tail  races,  and  graduated  so  that 
their  zero  points  will  be  at  some  datum  below  the  tail-race 
level.  During  the  test  of  a  wheel  each  gauge  is  read  by  an 
observer  at  stated  intervals,  and  the  difference  of  the  readings 
gives  the  head  h.  In  some  cases  it  is  possible  to  have  a  float- 
ing gauge  on  the  lower  level,  the  graduated  rod  of  which  is 
placed  alongside  of  a  glass  tube  that  communicates  with  the 
upper  level ;  the  head  h  is  then  directly  read  by  noting  the 
point  of  the  graduation  which  coincides  with  the  water  surface 
in  the  tube.  This  device  requires  but  one  observer,  while  the 
former  requires  two ;  but  it  is  usually  not  the  cheapest  arrange- 
ment unless  a  large  number  of  observations  are  to  be  taken. 

When  water  is  delivered  through  a  nozzle  or  pipe  to  a  hy- 
draulic motor  the  head  which  is  to  be  determined  for  ascertain- 
ing the  efficiency  of  the  motor  is  not  the  total  'fall,  since  a  large 
part  of  that  may  be  lost  in  friction  in  the  pipe,  but  is  merely 


ART.  122.]  MEASUREMENT  OF   THE  HEAD.  289 

V* 

the   velocity-head  —  of  the  issuing  jet.     The  value  of  v  is 

known  when  the  discharge  q  and  .the  area  of  the  cross-section 
of  the  stream  have  been  determined,  and 


2g  2gC 

In  the  same  manner  when  a  stream  flows  in  a  channel  against 
the  vanes  of  an  undershot  wheel  the  effective  head  is  the 
velocity-head,  and  the  theoretic  energy  is 


2g        2ga 

If,  however,  the  water  in  passing  through  the  wheel  falls  a 
distance  //0  below  the  mouth  of  the  nozzle,  then  the  effective 
head  is 

*»_+^ 

In  order  to  fully  utilize  the  fall  7/0  it  is  plain  that  the  wheel 
should  be  placed  as  near  the  level  of  the  tail  race  as  possible. 

When  water  enters  upon  a  wheel  through  an  orifice  which  is 
controlled  by  a  gate,  losses  of  head  will  result,  which  can  be 
estimated  by  the  rules  of  Chapters  IV  and  V.  If  this  orifice  is 
in  the  head  race  the  loss  of  head  should  be  subtracted  from  the 
total  head  in  order  to  obtain  the  h  which  really  acts  upon  the 
wheel.  But  if  the  regulating  gates  are  a  part  of  the  wheel 
itself,  as  is  the  case  in  a  turbine,  the  loss  of  head  should  not  be 
subtracted,  because  it  is  properly  chargeable  to  the  construc- 
tion of  the  wheel,  and  not  to  the  arrangements  which  furnish 
the  supply  of  water.  In  any  event  that  head  h  should  be 
determined  which  is  to  be  used  in  the  subsequent  discussions  : 
if  the  efficiency  of  the  fall  is  desired,  the  total  available  head  is 
required  ;  if  the  efficiency  of  the  motor,  that  effective  head  is  to 
be  found  which  acts  directly  upon  it  (Art.  120). 


2QO  MEASUREMENT  OF    WATER  POWER.  [CHAP.  X. 

Prob.  148.  A  pressure  gauge  at  the  entrance  of  a  nozzle 
registers  1 16  pounds  per  square  inch,  and  the  coefficient  of 
velocity  of  the  nozzle  is  0.98.  Compute  the  effective  velocity- 
head  of  the  issuing  jet. 

ARTICLE  123.  MEASUREMENT  OF  EFFECTIVE  POWER. 

The  effective  work  and  horse-power  delivered  by  a  water- 
wheel  or  hydraulic  motor  is  often  required  to  be  measured. 
Water-power  may  be  sold  by  means  of  the  weight  W,  or  quan- 
tity q,  furnished  under  a  certain  head,  leaving  the  consumer  to 
provide  his  own  motor ;  or  it  may  be  sold  directly  by  the  num- 
ber of  horse-power.  In  either  case  tests  must  be  made  from 
time  to  time  in  order  to  insure  that  the  quantity  contracted 
for  is  actually  delivered  and  is  not  exceeded.  It  is  also  fre- 
quently required  to  measure  effective  work  in  order  to  ascertain 
the  power  and  efficiency  of  the  motor,  either  because  the  party 
who  buys  it  has  bargained  for  a  certain  power  and  efficiency, 
or  because  it  is  desirable  to  know  exactly  what  the  motor  is 
doing  in  order  to  improve  if  possible  its  performance. 

The  effective  work  of  a  motor  might  be  measured  if  it  could 
be  used  to  operate  a  pump  in  which  are  no  losses  of  any  kind. 
This  pump  might  raise  the  same  water  that  drives  the  motor 
through  a  vertical  height  h^  ;  then  the  effective  work  per  second 
would  be  Wkl ,  and  the  efficiency  of  the  motor  would  be 

k  W/L          //, 


e  — _ 


K~  Wh  ~  h  ' 

It  is  needless,  however,  to  say  that  such  a  pump  is  purely  im- 
aginary. 

A  method  in  which  the  effective  work  of  a  small  motor  may 
be  measured  is  to  compel  it  to  exert  all  its  power  in  lifting  a 
weight.  For  this  purpose  the  weight  may  be  attached  to  a 
cord  which  is  fastened  to  the  horizontal  axis  of  the  motor,  and 
around  which  it  winds  as  the  shaft  revolves.  The  wheel  then 


ART.  123.]     MEASUREMENT  OF  EFFECTIVE  POWER,  29  1 

expends  all  its  power  in  lifting  this  weight  Wl  through  the 
height  kl  in  tl  seconds,  and  the  work  performed  per  second 
then  is 


This  method,  although  practicable,  is  usually  cumbersome  in 
actual  use,  on  account  of  the  difficulty  of  determining  tl  with 
precision,  since  the  height  hl  which  can  be  secured  is  generally 
small. 

The  usual  way  of  measuring  the  effective  power  is  by  means 
of  the  friction  brake  or  power  dynamometer,  which  is  described 
in  the  next  article.  By  this  method  the  effective  work  per  second 
k  is  readily  determined,  and  then  the  power  of  the  motor  is 

k 
hp  =  --  =  O.ooi8l8£, 

550 

and  its  efficiency  is  found  by  dividing  this  by  the  theoretic 
power. 

The  test  of  a  hydraulic  motor  has  for  its  object  :  first,  the 
determination  of  the  effective  energy  and  power  ;  secondly,  the 
determination  of  its  efficiency  ;  and  third,  the  determination  of 
the  speed  which  gives  the  greatest  power  and  efficiency.  If 
the  wheel  be  still,  there  is  no  power;  if  it  be  revolving  very  fast, 
the  water  is  flowing  through  it  so  as  to  change  but  little  of  its 
energy  into  work  :  and  in  all  cases  there  is  found  a  certain 
speed  which  gives  the  maximum  power  and  efficiency.  To 
execute  these  tests,  it  is  not  at  all  necessary  to  know  how  the 
motor  is  constructed  or  the  principle  of  its  action,  although 
such  knowledge  is  very  valuable,  and  is  in  fact  indispensable, 
in  order  to  enable  the  engineer  to  suggest  methods  by  which 
its  operation  may  be  improved. 

Prob.  149.  What  is  the  horse-power  of  a  motor  which  in 
75.5  seconds  lifts  a  weight  of  320  pounds  through  a  vertical 
height  of  42  feet  ? 


2Q2 


MEASUREMENT  OF    WATER  POWER.          [CHAP.  X. 


ARTICLE  124.  THE  FRICTION  BRAKE,  OR  POWER 
DYNAMOMETER. 

The  effective  work  k  performed  by  a  hydraulic  motor  is 
measured  by  an  apparatus,  invented  by  PRONY,  called  the  fric- 
tion brake.  In  Fig.  80  is  illustrated  a  simple  method  of  apply- 
ing it  to  a  vertical  shaft,  the 
upper  diagram  being  a  plan  and 
the  lower  an  elevation.  Upon 
the  vertical  shaft  is  a  fixed 
pulley,  and  against  this  are  seen 
two  rectangular  pieces  of  wood 
hollowed  so  as  to  fit  it,  and  con- 
nected by  two  bolts.  By  turn- 
ing the  nuts  on  these  bolts  while 
the  pulley  is  revolving,  the  fric- 
tion can  be  increased  at  pleas- 
ure, even  so  as  to  stop  the  mo- 
tion ;  around  these  bolts  between 
the  blocks  are  two  spiral  springs 
(not  shown  in  the  diagram)  which  press  the  blocks  outward 
when  the  nuts  are  loosened.  To  one  of  these  blocks  is  attached 
a  cord  which  runs  horizontally  to  a  small  movable  pulley  over 
which  it  passes,  and  supports  a  scale  pan  in  which  weights  are 
placed.  This  cord  runs  in  a  direction  opposite  to  the  motion 
of  the  shaft,  so  that  when  the  brake  is  tightened  it  is  prevented 
from  revolving  by  the  tension  caused  by  the  weights.  The 
direction  of  the  cord  in  the  horizontal  plane  must  be  such 
that  the  perpendicular  let  fall  upon  it  from  the  centre  of  trie 
shaft,  or  its  lever  arm,  is  constant ;  this  can  be  effected  by 
keeping  the  small  pointer  on  the  brake  at  a  fixed  mark  estab- 
lished for  that  purpose. 

To  measure  the  power  of  the  wheel,  the  shaft  is  disconnected 
from  the   machinery   which  it   usually  runs,  and  allowed   to 


FIG.  80. 


ART.  124.]  THE  FRICTION  BRAKE.  293 

» 

revolve,  transforming  all  its  work  into  heat  by  the  friction  be- 
tween the  revolving  pulley  and  the  brake  which  is  kept  station- 
ary by  tightening  the  nuts,  and-  at  the  same  time  placing 
sufficient  weights  in  the  scale  pan  to  hold  the  pointer  at  the 
fixed  mark.  Let  n  be  the  number  of  revolutions  per  second  as 
determined  by  a  counter  attached  to  the  shaft,  P  the  tension 
in  the  cord  which  is  equal  to  the  weight  of  the  scale  pan  and 
its  loads,  /  the  lever  arm  of  this  tension  with  respect  to  the 
centre  of  the  shaft,  r  the  radius  of  the  pulley,  and  F  the  total 
force  of  friction  between  the  pulley  and  the  brake.  Now  in 
one  revolution  the  force  F  is  overcome  through  the  distance 
2ttr,  and  in  n  revolutions  through  the  distance  2nrn.  Hence 
the  effective  work  done  by  the  wheel  in  one  second  is 

k  =  F  .  2nrn  =  2nn  .  Fr. 

The  force  F  acting  with  the  lever-arm  r  is  exactly  balanced  by 
the  force  P  acting  with  the  lever-arm  /;  accordingly, 

Fr  =  Pl-, 
and  hence  the  effective  work  per  second  is 

k  =  27tnPl, 
and  the  effective  horse-power  is 

27tnPl 
hp  —  -77—  =  o.o  I  i42nPl.     ....     (8 1) 

As  the  number  of  revolutions  in  one  second  cannot  be  accurately 
read,  it  is  usual  to  record  the  counter  readings  every  minute  or 
half  minute  ;  if  N  be  the  number  of  revolutions  per  minute, 

27tNPl 

hp  =.-         --  =  o.oooiao4/v7Y.   .     .     .     (81)' 
33  ooo 

It  is  seen  that  this  method  is  independent  of  the  radius  of  the 
pulley,  which  may  be  of  any  convenient  size  ;  for  a  small  motor 
the  brake  may  be  clamped  directly  upon  the  shaft,  but  for  a 
large  one  a  pulley  of  considerable  size  is  needed,  and  a  special 
arrangement  of  levers  is  used  instead  of  a  cord. 

The  efficiency  of  the  motor  is  now  found  by  dividing  the 


294  MEASUREMENT  OF    WATER   POWER,  [CHAP.  X. 

effective  work  by  the  theoretic  energy,  or  the  effective  power 
by  the  theoretic  power  ;  thus  : 

k        hp        ^     nPl 


This  same  formula  applies  if  the  number  of  revolutions  be  per 
minute,  provided  that  W  be  the  weight  of  water  which  flows 
through  the  wheel  per  minute. 

The  power  measured  by  the  friction  brake  is  that  delivered 
at  the  circumference  of  the  pulley,  and  does  not  include  that 
power  which  is  required  to  overcome  the  friction  of  the  shaft 
upon  its  bearings.  The  shaft  or  axis  of  every  water  wheel 
must  have  at  least  two  bearings,  the  friction  of  which  consumes 
probably  about  2  or  3  per  cent  of  the  power.  The  hydraulic 
efficiency  of  the  wheel,  regarded  as  a  user  of  water,  is  hence  2 
or  3  per  cent  greater  than  the  value  of  e  as  given  by  (82). 

There  are  in  use  various  forms  and  varieties  of  the  friction 
brake,  but  they  all  act  upon  the  principle  and  in  the  manner 
above  described.  For  large  wheels  they  are  made  of  iron,  and 
almost  completely  encircle  the  pulley;  while  a  special  arrange- 
ment of  levers  is  used  to  lift  the  large  weight  P.*  If  the 
work  transformed  into  friction  be  large,  both  the  brake  and  the 
pulley  may  become  very  hot,  to  prevent  which  a  stream  of  cool 
water  is  allowed  to  flow  upon  them.  To  insure  steadiness  of 
motion  it  is  well  that  the  surface  of  the  pulley  should  be  lubri- 
cated, which  for  a  wooden  brake  is  well  done  by  the  use  of 
soap. 

Prob.  150.  Find  the  power  and  efficiency  of  a  motor  when 
the  theoretic  energy  is  1.38  horse-power,  which  makes  670  revo- 
lutions per  minute,  the  weight  on  the  brake  being  2  pounds 
14  ounces  and  its  lever  arm  1.33  feet. 

*  A  paper  by  THURSTON  in  Transactions  of  American  Society  of  Mechanical 
Engineers  1886,  vol.  viii.,  gives  detailed  descriptions  and  illustrations  of  the 
testing  apparatus  at  Holyoke,  Mass. 


ART.  125.]  TEST  OF  A    SMALL  MOTOR.  295 

ARTICLE  125.  TEST  OF  A  SMALL  MOTOR. 

The  following  description  and  notes  of  a  test  of  the  6-inch 
Eureka  turbine  in  the  hydraulic  laboratory  of  Lehigh  Univer- 
sity may  serve  to  exemplify  the  preceding  method  of  determin- 
ing the  effective  power  and  efficiency.  The  water  was  deliv- 
ered over  a  weir  from  which  it  ran  into  a  vertical  penstock 
15.98  square  feet  in  horizontal  cross-section.  This  plan  of  hav- 
ing the  weir  above  the  wheel  is  in  general  not  a  good  one,  since 
it  is  then  difficult  to  maintain  a  constant  head  in  the  penstock  ; 
and  it  was  adopted  in  this  case  on  account  of  the  lack  of  room 
below  the  wheel,  and  for  other  reasons  which  need  not  here  be 
explained,  as  they  are  not  related  to  the  question  in  hand.  The 
weir  is  briefly  described  in  Art.  52,  and  the  depths  on  its  crest 
were  determined  by  a  hook  gauge  reading  to  thousandths  of  a 
foot.  When  a  constant  head  is  maintained  in  the  penstock 
the  quantity  of  water  flowing  through  the  wheel  is  the  same  as 
that  passing  the  weir  ;  if,  however,  the  head  in  the  penstock 
falls  x  feet  per  minute,  the  flow  Q  through  the  wheel  in  cubic 
feet  per  minute  is 


in  which  q  is  the  flow  per  second  through  the  weir,  as  com- 
puted by  the  methods  of  Chapter  V.  As  the  supply  of  water 
was  very  limited  the  wheel  could  not  be  run  to  its  full  capacity. 
There  was  no  leakage  from  the  penstock,  and  the  slight  leak- 
age through  the  gate  of  the  turbine  is  properly  included  in  the 
value  of  q,  since  it  assists  in  running  the  wheel. 

The  level  of  water  in  the  penstock  above  the  wheel  was 
read  upon  a  head  gauge  consisting  of  a  glass  tube  behind  which 
a  graduated  scale  was  fixed,  the  zero  of  which  was  a  little  above 
the  water  level  in  the  tail  race.  The  latter  level  was  read  upon 
a  fixed  graduated  scale  having  its  zero  in  the  same  horizontal 
plane  as  the  first  ;  these  readings  were  hence  essentially  nega- 


296 


MEASUREMENT  OF    WATER  POWER.          [CHAP.  X. 


tive.     The  head  upon  the  wheel  is  then  found  by  adding  the 
readings  of  the  two  gauges. 

The  vertical  shaft  of  the  turbine,  being  about  15  feet  long, 
was  supported  by  four  bearings,  and  to  a  small  pulley  upon  its 
upper  end  was  attached  the  friction  dynamometer,  as  described 
in  the  last  article.  The  number  of  revolutions  was  read  from 
a  counter  placed  in  the  top  of  this  shaft.  The  observations 
were  taken  at  minute  intervals,  electric  bells  giving  the  signals, 
so  that  precisely  at  the  beginning  of  each  minute  simultaneous 
readings  were  taken  by  observers  at  the  weir,  at  the  head  gauge, 
at  the  tail  gauge,  and  at  the  counter,  the  operator  at  the  brake 
continually  keeping  it  in  equilibrium  with  the  resisting  weight 
in  the  scale  pan  by  slightly  tightening  and  loosening  the  nuts 
as  required.  The  following  table  gives  the  notes  of  four  sets, 

TABLE   XXIII.    TEST   OF   A   6-INCH    EUREKA   TURBINE. 


Time  on 

Depth 

Penstock 

Tail-race 

Revolu- 

Weight 

April  13, 

1888. 

on  Weir 
Crest. 

Gauge. 

Gauge. 

tions  in 
One 

on 
Brake. 

Remarks. 

Feet. 

Feet. 

Feet. 

Minute. 

Pounds. 

3h     i7m 

0.288 

11.25 

—  O.2I 

2-5 

18 

0.289 

11.17 

0.20 

625 

tt 

Length  of  weir, 

19 

0.289 

11.13 

0.21 

**^3 

tt 

b  =  1.909  feet. 

635 

20 

0.288 

II.  IO 

O.2I 

Length    of   lever- 

3h       22m 

0.287 

10.  Si 

—  O.2O 

535 

3-0 

arm  on  brake, 
/=  1.431  feet. 

23 

0.287 

10.69 

O.2O 

" 

24 

0.287 

10.62 

0.21 

540 

535 

v 

Gate  of  wheel  f  ths 
open  during  all  ex- 

25 

0.286 

10.57 

O.2I 

" 

periments. 

3h       27m 
28 

0.288 
0.288 

10.64 
10.72 

-0.23 
0.22 

600 
600 

2-5 

Temperature     of 
the  water  not  taken. 

29 

0.291 

10.80 

0.21 

" 

615 

30 

0.290 

10.90 

0.20 

3h     32m 

0.290 

10.72 

—  O.2O 

3-5 

445 

33 

0.291 

10.69 

0.20 

" 

440 

5 

34 

0.291 

10.66 

O.2O 

" 

440 

I 

35 

0.292 

10.64 

0.20 

ART.  125.] 


TEST  OF  A    SMALL   MOTOR. 


297 


each  lasting  three  minutes,  the  weight  in  the  scale  pan  being 
different  in  each.  In  the  intervals  between  the  sets  the  wheel 
was  kept  running  and  observations  were  regularly  taken ;  but 
they  are  not  used,  owing  to  the  disturbance  of  the  permanent 
motion  consequent  upon  changing  the  weight  in  the  scale-pan. 

The  following  table  gives  the  results  of  the  computations 
made  from  the  above  notes  for  each   minute   interval.     The 
second  column  is  derived  from  formula  (33)  of  Art.  53,  using 
TABLE   XXIV.    RESULTS   OF   TEST   OF   A   6-INCH    TURBINE. 


Interval 
of 
Time. 

Discharge 
over  Weir. 

Cub.  Feet 
per 
Minute. 

Fall  in 
Penstock. 

Feet. 

Flow 
through 
Wheel. 

Cub.  Feet 
per 
Minute. 

Head  on 
Wheel. 

Feet. 

Theoretic 
Horse- 
power of 
the 
Water. 

Effective 
Horse- 
power of 
the 
Wheel. 

Efficiency 
of  the 

Wheel. 

Per  Cent. 

I7m  to  i8m 

53.49 

+  0.08 

59-77 

11.41 

1.290 

0-433 

33-6 

18     to   19 

58.66 

+0.04 

59-30 

11.36 

1.274 

0.426 

33-4 

19     to   20 

58.49 

+  0.03 

58.97 

11.32 

1.262 

0.433 

34-3 

22m    tO    23m 

58.05 

+  0.13 

60.13 

10.95 

1.245 

0-437 

35-1 

23     to  24 

58.05 

+  0.07 

59-J7 

10.86 

1.  215 

0.441 

36.3 

24     to  25 

57-88 

+  0.05 

58.68 

10.80 

1.198 

0-437 

36.5 

27m  to  28m 

58.36 

—  o.oS 

57.08 

10.91 

I-I75 

0.409 

34-7 

28     to   29 

58.51 

—  0.08 

57.23 

10.97 

1.187 

0.409 

34-4 

29     to  30 

59.10 

—  O.  IO 

57-50 

1  1.  06 

1.203 

0.419 

34-8 

32m    to    33m 

59.10 

+0.03 

59-58 

10.90 

1.228 

0.424 

34-5 

33     to  34 

59-26 

+0.03 

59-74 

10.87 

1.228 

0.420 

34-2 

34     to  35 

59-41 

-(-0.02 

59-73 

10.85 

1.226 

0.420 

34-3 

the  coefficient  corresponding  to  the  given  length  of  weir  and 
depth  on  crest.  The  third  column  is  obtained  by  taking  the 
differences  of  the  observed  readings  of  the  penstock  head 
gauge.  The  fourth  column  is  the  value  of  Q  found  as  above 
explained.  The  fifth  column  is  the  mean  head  h  on  the  wheel 
during  the  minute,  as  derived  from  the  observed  readings  of 
head  and  tail  gauge.  The  sixth  column  is  found  by  formula 
(80),  using  for  W  its  value ' -fcwQ,  in  which  w\s  taken  at  62.4 


298  MEASUREMENT  OF    WATER  POWER.          [CHAP.  X. 

pounds  per  cubic  foot.  The  seventh  column  is  computed  from 
formula  (Si)7;  and  the  last  column  is  found  by  dividing  the 
numbers  in  the  seventh  by  those  in  the  sixth  column. 

These  results  show  that  the  mean  efficiency  of  the  wheel 
and  shaft  under  the  conditions  stated  was  about  35  per  cent; 
also,  that  the  efficiency  in  the  second  set  is  the  highest  and 
that  in  the  first  is  the  lowest.  The  following  recapitulation  of 
the  means  for  each  set  show  that  the  reason  for  the  variation 
in  efficiency  is  the  variation  in  speed,  and  it  is  to  be  concluded 

N.  h.  Q.  e. 

ist  set,          632         11.36         59.31         33.8 

3d  set,          605         10.98         57.27         34.6 

2d  set,          536         10.87         59-33         36-O 

4th  set,         441          10.87         59-68         34.3 

that  with  a  head  of  about  1 1  feet  this  wheel  at  three-fourths 

gate  has  a  maximum  efficiency  of  36.0  per  cent  when  running 

at  about  535  revolutions  perrqinute.     If  four  points  be  plotted, 

taking  the  values  of  N  as  abscissas  and  those  of  e  as  ordinates, 

and  a  curve  be  drawn  through  them,  it  will  be  seen  that  quite 

material  variations  in  the  speed  may  occur  without  sensibly 

affecting  the  efficiency;  thus  N  may  range  from  475  to  575 

revolutions  per  minute  without  making  e  lower  than  0.35. 

Prob.  151.  Compute,  using  four-figure  logarithms,  the  re- 
sults in  the  last  three  columns  of  the  above  table. 

ARTICLE  126.  LOWELL  AND  HOLYOKE  TESTS. 

The  work  of  FRANCIS  on  the  experiments  made  by  him  at 
Lowell  will  always  be  a  classic  in  American  hydraulic  litera- 
ture, for  the  methods  therein  developed  for  measuring  the 
theoretic  power  of  a  water-fall,  and  the  effective  power  utilized 
by  the  wheel,  are  models  of  careful  and  precise  experimenta- 
tion.* In  determining  the  speed  of  the  wheel  he  used  a  method 

*  Lowell  Hydraulic  Experiments,  ist  Edition,  1855  ;  4th,  1883. 


ART.  126.]  LOWELL  AND  HOLYOKE    TESTS.  299 

somewhat  different  from  that  above  explained,  namely,  the 
counter  attached  to  the  shaft  was  connected  with  a  bell  which 
struck  at  the  completion  of  every.  50  revolutions ;  the  observer 
at  the  counter  had  then  only  to  keep  his  eye  upon  the  watch, 
and  to  note  the  time  at  certain  designated  intervals — say  at 
every  sixth  stroke  of  the  bell.  The  number  of  revolutions  per 
second  was  then  obtained  by  dividing  the  number  of  revolu- 
tions in  the  interval  by  the  number  of  seconds,  as  determined 
by  the  watch.  This  method  requires  a  stop-watch  in  order  to 
do  good  work,  unless  the  observer  be  very  experienced,  as  an 
error  of  one  second  in  an  interval  of  one  minute  amounts  to 
1.7  per  cent. 

The  Holyoke  Water  Power  Company  has  a  permanent 
flume  for  testing  turbines  arranged  with  a  weir  which  can  be 
varied  up  to  lengths  of  20  feet,  so  as  to  test  the  largest  wheels 
which  are  constructed.  As  the  expense  of  fitting  up  the  ap- 
paratus for  testing  a  large  turbine  at  the  place  where  it  is  to 
be  used  is  often  great,  it  is  sometimes  required  in  contracts 
that  the  wheel  shall  be  sent  to  Holyoke  to  be  tested,  that  be- 
ing the  only  place  in  the  United  States  where  a  special  outfit 
for  such  work  exists.  The  wheel  is  mounted  in  the  testing 
flume,  and  there,  by  the  methods  explained  in  the  preceding 
articles,  it  is  run  at  different  speeds  in  order  to  determine  the 
speed  which  gives  the  maximum  efficiency  as  well  as  the  effec- 
tive power  developed  at  each  speed.  As  the  efficiency  of  a 
turbine  varies  greatly  with  the  position  of  the  gate  which  ad- 
mits the  water  to  it,  tests  are  made  with  the  gate  fully  opened 
and  at  various  partial  openings.  The  results  thus  obtained  are 
not  only  valuable  in  furnishing  full  information  concerning  the 
effective  power  and  efficiency  of  the  wheel,  but  they  also  con- 
vert the  turbine  into  a  water  meter,  so  that  when  running  under 
the  same  head  as  during  the  tests  the  quantity  of  water  which 
passes  through  it  can  at  any  time  be  approximately  ascer- 
tained. 


300 


MEASUREMENT  OF    WATER  POWER.          [CHAP.  X. 


The  following  table  gives  a  report  of  the  Holyoke  Water 
Company  of  the  test  of  an  8o-inch  outward-flow  BOYDEN  tur- 
bine made  September  22,  1885.*  The  headings  of  the  several 

TABLE  XXV.    TEST   OF   AN    So-INCH    BOYDEN   TURBINE. 


Proportional  Part 

of 

the  full 

Quantity 

Number 

Discharge 

Head 

Dura- 

Revolu- 

of 
Water 

Power 

Efficien- 

of the 
Experi- 
ment. 

the  full 
Opening 
of  the 
Speed- 

Wheef; 
being  the 
Discharge 
at  full 

acting 
on  the 
Wheel. 

tion  of 
the  Ex- 
peri- 
ment. 

tions 
of  the 
Wheel. 

dis- 
charged 
by  the 
Wheel. 

developed 
by  the 

Wheel. 

cy  of 
the 
Wheel. 

gate. 

Gate 

when  giv- 

Cub. Ft. 

ing  best 
Efficiency 

Feet. 

Min. 

Per  Min. 

per 
Second. 

h.  p. 

Per  Cent 

21 

I.OOO 

0.992 

17.16 

5 

63.50 

117.01 

172.57 

75-85 

20 

« 

.000 

17.27 

5 

70.00 

118.37 

177.41 

76.60 

19 

" 

.008 

17-33 

3 

75-00 

H9-53 

178.63 

76.11 

1  8 

" 

.O2O 

17.34 

3 

80.00 

121.15 

178.32 

74-92 

17 

•* 

.036 

17.21 

2 

86.00 

122.41 

178.57 

74-81 

16 

" 

.056 

17.21 

5 

93-20 

124.74 

176.44 

72.54 

15 

" 

.082 

17.19 

3 

IOO.OO 

127-73 

167.94 

67-51 

14 

0-753 

0.923 

17.26 

4 

65.00 

IOg.22 

148.86 

69.69 

13 

4  ' 

0.931 

17-35 

4 

71.00 

110.42 

151.76 

69.91 

12 

" 

0.944 

17.33 

3 

77.17 

111.94 

153.16 

69.68 

II 

<  < 

0-957 

17-34 

3 

82.83 

II3-52 

I5L75 

68.04 

10 

« 

0.986 

17-34 

3 

93-33 

116.98 

151.04 

65.72 

9 

11 

0.999 

17.27 

4 

97-75 

118.24 

140.29 

60.63 

8 

C  ( 

I.OlS 

17.23 

4 

104.50 

120.36 

130.83 

55-68 

33 

0.609 

0.849 

17.64 

4 

65.00 

101.60 

130.99 

64.51 

32 

'« 

0.861 

17-57 

3 

71  .00 

102.78 

130.08 

63-58 

3i 

" 

0.876 

17-53 

4 

78.00 

104.45 

I28.6I 

62.00 

30 

" 

0.892 

17-45 

4 

84.75 

106.  19 

124.22 

59.16 

7 

0.438 

0.706 

17.68 

3 

64.00 

84.56 

84-03 

49.61 

6 

0.716 

17.69 

4 

69.25 

85-74 

82.47 

47-99 

5 

" 

0.723 

17.69 

4 

74-75 

86.57 

79-89 

46.04 

4 

M 

0.731 

17.66 

4 

79.87 

87.54 

75.60 

43.16 

3 

11 

0.746 

17.62 

4 

86.50 

89-23 

68.67 

38.55 

2 

" 

0.762 

17.64 

3 

94-33 

91.12 

57.61    j    31.63 

I 

" 

0.773 

17.61 

4 

100.50 

92.36 

46.03    |    24.98 

27 

0.310 

0-555 

18.03 

4 

6i.75 

67.11 

43-37   1    31.63 

26 

II 

0.570 

iS.OI 

4 

69.25 

68.87 

40.18 

28.59 

25 

11 

0.584 

18.02 

3 

77.00 

70  59 

35.27 

24.47 

28 

11 

0-597 

18.23 

3 

85.00 

72.62 

28.55 

19.03 

29 

"• 

0.608 

18.13 

3 

90.67 

73-74 

19.38 

12.79 

24 

0.200 

0.401 

18.17 

3 

54-33 

48.68 

18.25 

18.21 

23 

" 

0.409 

18.11 

3 

61.67 

49.52        15.06 

14.83 

22 

0.413 

18.07 

4 

66.50 

50.03 

12.18 

11.89 

*  Kindly  furnished  by  Mr.  CLEMENS  HERSCHEL,  Hydraulic  Engineer  of  the 
Holyoke  Water  Power  Company. 


ART.  126.]  LOWELL  AND  HOLYOKE    TESTS.  3OI 

columns  sufficiently  designate  their  meaning,  and  only  a  few 
words  need  be  added  explanatory  of  the  method  by  which 
they  were  obtained.  The  numbers  in  the  second  column  were 
found  by  actual  measurement  of  the  clear  space  beneath  the 
gate,  the  space  at  full  gate  being  called  unity ;  those  in  the 
fourth  column  are  derived  from  the  head  and  tail  race  gauges ; 
those  in  the  sixth  column  by  dividing  the  total  number  of  revo- 
lutions during  the  experiment  by  its  length  in  minutes ;  those 
in  the  seventh  by  the  measurement  of  the  water  over  the  weir ; 
those  in  the  eighth  from  the  friction  dynamometer  by  the  use 
of  formula  (Si)7;  and  those  in  the  last  column  were  computed 
by  (82).  The  quantities  in  the  third  column  result  from  the 
division  of  those  in  the  seventh  by  118.37,  that  being  the  dis- 
charge at  full  gate  for  maximum  efficiency ;  it  is  seen  from 
these  that  the  discharge  depends  not  only  upon  the  head,  but 
on  the  velocity  of  the  wheel,  and  that  it  always  increases  when 
the  speed  increases.* 

The  following  data  regarding  the  dimensions  of  this  wheel 
are  here  also  noted,  as  it  may  be  necessary  to  refer  to  it  again 
when  the  subject  of  turbines  is  discussed : 

Outer  radius  of  wheel  r^  =  3.3167  feet ; 

Inner  radius  of  wheel  r    =  2.6630  feet ; 

Outer  radius  of  guide  case  r0  =  2.5911  feet; 

Outer  depth  of  buckets  d^  =  0.722  feet; 

Inner  depth  of  buckets  d  =  0.741  feet; 

Outer  area  .of  buckets  al  =  4.61  square  feet; 

Inner  area  of  buckets  a   =12.12  square  feet ; 

Outer  area  of  guide  orifices  a0  =  4.76  square  feet ; 

Exit  angle  of  buckets  ft  =  13.5  degrees; 

Entrance  angle  of  buckets  0  =  90  degrees  ; 

Entrance  angle  of  guides  a  =  24  degrees  ; 
Number  of  buckets,  52.        Number  of  guides,  32. 

*  For  further  examples  of  tests  at  Holyoke  see  a  paper  by  THURSTON  in 
Transactions  American  Society  Mechanical  Engineers,  vol.  viii.  p.  359. 


302  MEASUREMENT  OF    WATER  POWER.  [CHAP.  X. 

Prob.  152.  In  experiment  22  on  the  outward-flow  turbine 
only  about  12  per  cent  of  the  theoretic  power  is  utilized.  How 
is  the  remaining  88  per  cent  expended? 

ARTICLE  127.  WATER  POWER. 

In  1880  there  was  employed  in  the  United  States  a  total  of 
3  410  837  horse-power,  of  which  about  36  per  cent  was  derived 
from  water  and  about  64  per  cent  from  steam.  It  has  been 
estimated  that  the  rivers  of  the  United  States  can  furnish 
about  200  ooo  ooo  horse-powers,  so  that  the  possibilities  for  the 
future  are  almost  unlimited,  and  when  coal  becomes  high  in 
price  water  is  sure  to  take  the  place  of  steam. 

Water-power  is  often  sold  by  what  is  called  the  "  mill 
power,"  which  may  be  roughly  supposed  to  be  such  a  quantity 
as  the  average  mill  requires,  but  which  in  any  particular  case 
must  be  denned  by  a  certain  quantity  of  water  under  a  given 
head.  Thus  at  Lowell  the  mill  power  is  30  cubic  feet  per 
second  under  a  head  of  25  feet,  which  is  equivalent  to  85.2 
theoretic  horse-power.  At  Minneapolis  it  is  30  cubic  feet  per 
second  under  22  feet  head,  or  75  theoretic  horse-power.  At 
Holyoke  it  is  38  cubic  feet  per  second  under  20  feet  head,  or 
86.4  theoretic  horse-power.  This  seems  an  excellent  way  to 
measure  power  when  it  is  to  be  sold  or  rented,  as  the  head  in 
any  particular  instance  is  not  subject  to  much  variation  ;  or  if 
so  liable,  arrangements  must  be  adopted  for  keeping  it  nearly 
constant,  in  order  that  the  machinery  in  the  mill  may  be  run  at 
a  tolerably  uniform  rate  of  speed.  Thus  nothing  remains  for 
the  water  company  to  measure  except  the  water  used  by  the 
consumer.  The  latter  furnishes  his  own  motor,  and  is  hence 
interested  in  securing  one  of  high  efficiency,  that  he  may  derive 
the  greatest  power  from  the  water  for  which  he  pays.  The 
perfection  of  American  turbines  is  undoubtedly  largely  due  to 
this  method  of  selling  power,  and  the  consequent  desire  of  the 


ART.  127.]  WATER  POWER.  303 

mill  owners  to  limit  their  expenditure  of  water.  The  turbine 
itself  when  tested  and  rated  becomes  a  meter  by  which  the 
company  can  at  any  time  determine  the  quantity  of  water  that 
passes  through  it.  At  Holyoke  the  cost  of  one  mill  power  for 
16  hours  a  day  is  $300  per  annum.* 

The  available  power  of  natural  water-falls  is  very  great,  but 
it  is  probably  exceeded  by  that  which  can  be  derived  from  the 
tides  and  waves  of  the  ocean.  Twice  every  day,  under  the 
attraction  of  the  sun  and  moon,  an  immense  weight  of  water  is 
lifted,  and  it  is  theoretically  possible  to  derive  from  this  a 
power  due  to  its  weight  and  lift.  Continually  along  every 
ocean  beach  the  waves  dash  in  roar  and  foam,  and  energy  is 
wasted  in  heat  which  by  some  device  might  be  utilized  in 
power.  The  expense  of  deriving  power  from  these  sources  is 
indeed  greater  than  that  of  the  water  wheel  under  a  natural 
fall,  but  the  time  may  come  when  the  profit  will  exceed  the 
expense,  and  then  it  will  certainly  be  done.  Coal  and  wood 
and  oil  may  become  exhausted,  but  as  long  as  the  force  of 
gravitation  exists,  and  the  ocean  remains  upon  which  it  can  act, 
heat,  light,  and  power  can  be  generated  in  quantity  practically 
without  limit. 

Prob.  153.  Show  that  over  600  horse-power  is  wasted  in 
heat  for  every  square  mile  of  ocean  surface  where  the  rise  and 
fall  of  the  tide  is  3  feet. 

*  BRECKENRIDGE,  Journal  of  Engineering  Society  of  Lehigh  University, 
1887,  vol.  ii.  p.  34;  an  article  giving  a  detailed  account  of  the  water  power  at 
Holyoke. 


304        DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  XI. 


CHAPTER   XI. 
DYNAMIC    PRESSURE    OF    FLOWING    WATER. 

ARTICLE  128.  DEFINITIONS  AND  PRINCIPLES. 

The  pressures  exerted  by  moving  water  against  surfaces 
which  change  its  direction  or  check  its  velocity  are  called 
dynamic,  and  they  follow  very  different  laws  from  those  which 
govern  the  static  pressures  that  have  been  discussed  and  used  in 
the  preceding  chapters.  A  static  pressure  due  to  a  certain  head 
may  cause  a  jet  to  issue  from  an  orifice ;  but  this  jet  in  imping- 
ing upon  a  surface  may  cause  a  dynamic  pressure  less  than,  equal 
to,  or  greater  than  that  due  to  the  head.  A  static  pressure  at  a 
given  point  in  a  mass  of  water  is  exerted  with  equal  intensity 
in  all  directions ;  but  a  dynamic  pressure  is  exerted  in  different 
directions  with  different  intensities.  In  the  following  chapters 
the  words  static  and  dynamic  will  generally  be  prefixed  to  the 
word  pressure,  so  that  no  intellectual  confusion  may  result. 

The  dynamic  pressure  exerted  by  a  stream  flowing  with  a 
given  velocity  against  a  surface  at  rest  is  evidently  equal  to 
that  produced  when  the  surface  moves  in  still  water  with  the 
same  velocity.  This  principle  was  applied  in  Art.  109  in  rating 
the  current  meter,  whose  vanes  move  under  the  impulse  of  the 
impinging  water.  The  dynamic  pressure  exerted  upon  a  body 
by  a  flowing  stream  hence  depends  upon  the  velocity  of  the 
stream  and  surface. 

The  impulse  of  a  jet  or  stream  of  water  is  the  dynamic 
pressure  which  it  is  capable  of  producing  in  the  direction  of  its 
motion  when  its  velocity  is  entirely  destroyed  in  that  direction. 


ART.  128.]  DEFINITIONS  AND  PRINCIPLES.  305 

This  can  be  done  by  deflecting  the  jet  normally  sidewise  by  a 

fixed  surface  ;  if  the  surface  is  smooth,  so  that  no  energy  is  lost 

in  frictional  resistances,  the  actual  velocity  remains  unaltered, 

but  the  velocity  in    the  original  direction  has  been   rendered 

null.     In  Art.  32  it  is  proved  that  the  theoretic  force  of  im-   ^v  lf(( 

pulse  of  a  stream  of  cross-section  a  and  velocity  v  is 


F=  WV-  =  wq"-  =  2WH-,    .".      .     .     (83) 

in  which  W  and  q  are  the  weight  and  volume  delivered  per 

second,  and  w  is  the  weight  of  one  cubic  unit  of  water.     This 

equation  shows  that  the  dynamic  pressure  that  may  be  pro- 

duced by  impulse  is  equal  to  the  static  pressure  due  to  twice 

the  head  corresponding  to  the  velocity  v. 

It  would  then  be  expected  that  if  two 

equal  orifices    or  tubes  be  placed    ex- 

actly opposite,  as  in  Fig.  81,  and  a  loose 

plate  be  placed  vertically  against  one 

of  them,    that    the    dynamic   pressure  ^^^ 

upon  the  plate  caused  by  the  impulse  ~~— 

of   the  jet   issuing  from  A    under  the  FIG.  81. 

head  h  would   balance  the  static  pressure  caused  by  the  head 

2/j.     This  conclusion  has  been  confirmed  by  experiment,  when 

the  tube  A  has  a  smooth  inner  surface  and  rounded  inner  edges 

so  that  its  coefficient  of  discharge  is  unity. 

The  reaction  of  a  jet  or  stream  is  the  backward  dynamic  pres- 
sure, in  the  line  of  its  motion,  which  is  exerted  against  a  vessel  out 
of  which  it  issues,  or  against  a  surface  away  from  which  it  moves. 
This  is  equal  and  opposite  to  the  impulse,  and  the  equation 
above  given  expresses  its  value  and  the  laws  which  govern  it. 

The  expression  for  the  reaction  or  impulse  F  given  by  (83) 
may  be  also  proved  as  follows  :  The  definition  by  which  forces 


306        DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  Xk 

are  compared  with  each  other  is,  that  forces  are  proportional 
to  the  accelerations  which  they  can  produce.  The  weight  H 
if  allowed  to  fall  acquires  the  acceleration  g ;  the  force  F  which 
can  produce  the  acceleration  v  is  hence  related  to  W  and  g  by 
the  equation  F -^  v  —  W '-f-  g. 

Impulse  and  reaction  in  a  cross-section  of  a  stream  flowing 
with  constant  velocity  and  direction  are  forces  which  can  be 
exerted,  and  hence  like  energy  are  potential.  If  the  direction 
of  the  stream  be  changed  by  opposing  obstacles,  the  impulse 
and  reaction  produce  dynamic  pressure ;  if  in  making  this 
change  the  absolute  velocity  is  retarded,  energy  is  converted 
into  work.  Impulse  and  reaction  are  of  no  practical  value, 
except  in  so  far  as  the  resulting  dynamic  pressures  may  be 
utilized  for  the  production  of  work.  For  this  purpose  water  is 
made  to  impinge  upon  moving  vanes,  which  alter  both  its  direc- 
tion and  velocity,  thus  producing  a  dynamic  pressure  Pt  which 
overcomes  in  each  second  an  equal  resisting  force  through  the 
space  u.  The  work  done  per  second  is  then 

k  —  Pu. 

It  is  the  object  in  designing  a  hydraulic  motor  to  make  this 
work  as  large  as  possible,  and  for  this  purpose  the  most  advan- 
tageous values  of  Pand  u  are  to  be  selected. 

The  word  impact,  which  is  sometimes  erroneously  used  to 
mean  impulse  or  pressure,  properly  refers  to  those  cases  where 
energy  is  lost  through  changes  of  cross-section  (Art.  68),  or  in 
eddies  and  foam,  as  when  a  jet  impinges  into  water  or  upon  a 
rough  plane  surface.  When  work  is  to  be  utilized,  impact 
should  be  avoided  as  far  as  possible. 

Prob.  1 54.  If  a  jet  is  one  inch  in  diameter,  how  many  gallons 
per  second  must  it  deliver  in  order  that  its  impulse  may  be  100 
pounds? 


ART.  129.]  EXPERIMENTS  ON  IMPULSE  AND   REACTION.      3O/ 


ARTICLE  129.  EXPERIMENTS  ON  IMPULSE  AND  REACTION. 

In  Fig.  82  is  shown  a  simple  device  by  which  the  dynamic 
pressure  P  exerted  upon  a  surface  by  the  impulse  and  reaction 
of  a  jet  that  glides  over  it  can  be  directly  weighed.  It  consists 
merely  of  a  bent  lever  supported 
on  a  pivot  at  O,  and  having  a  plate 
A  attached  at  lower  end  of  the 
vertical  arm  upon  which  the  stream 
impinges,  while  weights  applied  at 
the  end  of  the  other  arm  meas- 
ure the  dynamic  pressure  produced 
by  the  impulse.  By  means  of  an 
apparatus  of  this  nature,  experiments  have  been  made  by 
BlDONE,  WE1SBACH,  and  others,  the  results  of  which  will  now 
be  stated. 

When  the  surface  upon  which  the  stream  impinges  is  a 
plane  normal  to  the  direction  of  the  stream,  as  shown  at  A, 
the  dynamic  pressure  P  closely  agrees  with  that  given  by  the 
theoretic  formula  for  F  in  the  last  article,  viz., 


FIG.  82. 


P=  W-  = 

g 


being  about  2  per  cent  greater  according  to  BlDONE,  and 
about  4  per  cent  less  according  to  WEISBACH.  The  actual 
value  of  P  was  found  to  vary  somewhat  with  the  size  of  the 
plate,  and  with  its  distance  from  the  end  of  the  tube  from  which 
the  jet  issued. 

When  the  surface  upon  which  the  stream  impinges  is  curved, 
as  at  B,  or  so  arranged  that  the  water  is  turned  backward  from 
the  surface,  the  value  of  the  dynamic  pressure  P  was  found  to 
be  always  greater  than  the  theoretic  value,  and  that  it  increased 


308        DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  XL 

the  greater  the  amount  of  backward  inclination.  When  a 
complete  reversal  of  the  original  direction  of  the  water  was 
obtained,  as  at  C,  it  was  found  that  P,  as  measured  by  the 
weights,  was  nearly  double  the  value  of  that  against  the  plane. 
This  is  explained  by  stating  that  as  long  as  the  direction  of  the 
flow  is  toward  the  surface  the  dynamic  pressure  of  its  impulse 
is  exerted  upon  it  ;  when  the  water  flows  backward  a\vay  from 
the  surface  the  dynamic  pressure  of  its  reaction  is  also  exerted 
upon  it.  The  sum  of  these  is 

P=F+F=2W-  =  4wa—, 


which  agrees  with  the  results  experimentally  obtained. 

An  experiment  by  MOROSI*  shows  clearly  that  the  dynamic 
pressure  against  a  surface  may  be  increased  still  further  by  the 
device  shown  in  Fig.  83,  where  the  stream  is  made  to  perform 
two  complete  reversals  upon  the  surface.     He  found  that  in 
this  case  the  value  of  the  dynamic  pressure  was 
3.32  times  as  great  as  that  against  a  plane,  or 
P  =  3.32  F,  whereas  theoretically  the  3.32  should 
be  4.     In  this  case,  as  in  those  preceding,  the 
water  in  passing  over  the  surface  loses  energy  in 
FIG.  83.  friction  and   foam,  so  that  its  velocity  is  dimin- 

ished, and  it  should  hence  be  expected  that  the  experimental 
values  of  the  dynamic  pressures  would  be  less  than  the  theo- 
retic values,  as  in  general  they  are  found  to  be. 

While  the  experiments  here  briefly  described  thoroughly 
confirm  the  results  of  theory,  they  further  show  it  is  the  change 
in  direction  of  the  velocity  when  in  contact  with  the  surface 
which  produces  the  dynamic  pressure.  If  the  stream  strikes 


*  RUHLMAN'S  Hydromechanik  (Hannover,  1879),  p.  586. 


ART.  129.]  EXPERIMENTS  ON  IMPULSE  AND  REACTION.      309 


normally  against  a  plane,  the  direction  of  its  velocity  is  changed 
90°,  and  this  is  the  same  as  the  entire  destruction  of  the 
velocity  in  its  original  direction,  so  that  the  dynamic  pressure  P 
should  agree  with  the  impulse  F.  This  important  principle  of 
change  in  direction  will  be  theoretically  exemplified  later. 

The  dynamic  pressure  produced  by  the  direct  reaction  of 
water  when  issuing  from  a  vessel  was  meas- 
ured by  EWART  with  the  apparatus  shown 
in  Fig.  84,  which  will  be  readily  understood 
without  a  detailed  description.  The  dis- 
cussion of  these  experiments  made  by  WET- 
BACK* shows  that  the  measured  values  of 
P  were  from  2  to  4  per  cent  less  than  the 
theoretic  value  F  as  given  by  (83),  so  that  in  FlG-  84- 

this  case  also  theory  and  observation  are  in  accordance. 

An  experiment  by  UNWlN,f  illustrated  in  Fig.  85,  is  very 
interesting,  as  it  perhaps  explains  more  clearly  than  formula 
(83)  why  it  is  that  the  dynamic  pressure  due  to  impulse  is 
double  the  static  pressure.  Two  ves- 
sels having  converging  tubes  of  equal 
size  were  placed  so  that  the  jet  from 
A  was  directed  exactly  into  B.  The 
head  in  A  was  kept  uniform  at  20^ 
inches,  when  it  was  found  that  the 
water  in  B  continued  to  rise  until  a  FlG- 8s> 

head  of  18  inches  was  reached.  All  the  water  admitted  into  A 
was  thus  lifted  in  B  by  the  impulse  of  the  jet,  with  a  loss  of  2-J 
inches  of  head,  which  was  caused  by  foam  and  friction.  If 
such  losses  could  be  entirely  avoided,  the  water  in  B  might  be 
raised  to  the  same  level  as  that  in  A.  In  the  case  shown  in 


*  Theoretical  Mechanics,  COXE'S  translation,  p.  1004. 
f  Encyclopaedia  Britannica,  gth  Edition,  vol.  xii.  p.  467. 


310        DYNAMIC  PRESSURE    OF  FLOWING    WATER.     [CHAP.  XI. 

the  figure  where  the  water  overflows  from  B,  the  impulse  of 
the  jet  has  not  only  to  overcome  the  static  pressure  due  to  the 
head  h,  but  also  to  furnish  the  dynamic  pressure  equivalent  to 
a  second  head  h  in  order  to  raise  the  water  through  that  height. 
But  the  level  in  B  can  never  rise  higher  than  in  A,  for  the  ve- 
locity-head of  the  jet  cannot  be  greater  than  that  of  the  static 
head  which  generates  it. 

Prob.  155.  In  Fig.  82  the  diameter  of  the  tube  is  I  inch, 
and  it  delivers  0.3  cubic  feet  per  second.  Compute  the  theo- 
retic dynamic  pressure  against  the  plane. 

ARTICLE  130.  SURFACES  AT  REST. 

Let  a  jet  of  water  whose  cross-section  is  a  impinge  in  per- 
manent flow  with  the  uniform  velocity  v  upon  a  surface  at  rest. 
Let  the  surface  be  smooth,  so  that  no  resisting  forces  of  fric- 
tion exist,  and  let  the  stream  be  prevented  from  spreading  lat- 
erally. The  water  then  passes  over  the  surface,  and  leaves  it 


FIG.  86. 

with  the  original  velocity  v,  producing  upon  it  a  dynamic  pres- 
sure whose  value  depends  upon  its  change  of  direction.  At  B 
in  Fig.  86  the  stream  is  deflected  normal  to  its  original  direc- 
tion, and  at  D  a  complete  reversal  is  effected,  Let  6  be  the 
angle  between  the  initial  and  final  directions,  as  shown.  It  is 
required  to  determine  the  dynamic  pressure  exerted  upon  the 
surface  in  the  same  direction  as  that  of  the  jet.  In  Fig.  86,  as 
in  those  that  follow,  the  stream  is  supposed  to  lie  in  a  horizon- 
tal plane,  so  that  no  acceleration  or  retardation  of  its  velocity 
will  be  produced  by  gravity. 


ART.  130.] 


SURFACES  AT  REST. 


The  stream  entering  upon  the  surface  exerts  its  impulse  F 
in  the  same  direction  as 
that  of  its  motion  ;  leaving 
the  surface  it  exerts  its  reac- 
tion F  in  opposite  direction 
to  that  of  its  motion.  Let 
P  be  the  dynamic  pressure 
thus  produced  in  the  direc- 
tion of  the  initial  motion,  Fl  the  component  of  the  reaction  F 
in  the  same  direction.  Then,  if  0  be  less  than  90°, 

P  =  F  —  F,  =  F(i  —  cos  0) ; 
and  if  0  be  greater  than  90°, 

p—F-\-Fl  =  ^+^cos(i8o°—  0)  =  F(i  —  cosfl). 

Both  cases  thus  give  the  same  result,  and  inserting  for  F  its 
value  as  given  by  (83)  ; 


P  =  (i  -  cos  0)W-  , 

<5 


(84) 


which  is  the  formula  for  the  dynamic  pressure  in  the  direction 
of  the  impinging  jet.  If  in  this  0  =  o°,  the  stream  glides  along 
the  surface  without  changing  its  direction,  and  P  becomes  zero  ; 
if  0  is  90°,  the  dynamic  pressure  is 


g 


and  if  0  becomes  180°  a  complete  reversal  of  direction  is  ob- 
tained, and 

P=2F=2W-. 
g 

These  theoretic  conclusions  agree  with  the  experimental  results 
described  in  the  last  article. 

The  resultant  dynamic  pressure  exerted  upon  the  surface  is 
found  by  combining  by  the  parallelogram  of  forces  the  impulse 


312         DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  XL 

F  and  the  equal  reaction  F.  In  Fig.  87  it  is  seen  that  this 
resultant  bisects  the  angle  180  —  9,  and  that  its  value  is 

P'  =  2Fcos£(iSo-  0)  =  2  sinJ0.fr-. 

o 

It  is  usually,  however,  more  important  to  ascertain  the  pres- 
sure in  a  given  direction  than  the  resultant. 
This  can  be  found  by  taking  the  component 
of  the  resultant  in  that  direction,  or  by  tak- 
ing the  algebraic  sum  of  the  components  of 
the  initial  impulse  and  the  final  reaction. 

To  find  the  dynamic  pressure  P  in  a  di- 
rection which  makes  an  angle   a  with  the 
entering  and  the  angle  6  with  the  depart- 
ing stream,  the  components  in  that  direction  are 

P,  =  F  cos  a,        P^  =  F  cos  0; 
and  the  algebraic  sum  of  these  is 

P  =  F(cos  a  —  cos  0)  =  (cos  a  —  cos  B)W- .    .     (84)' 

o 

This  becomes  equal  to  F  when  a  =  o  and  0  =  90°,  as  at  B  in 
Fig.  86,  and  also  when  a  =  90°  and  6  =  180°.  When  a  —  o° 
and  6  —  180°  the  entering  and  departing  streams  are  parallel, 
as  at  D  in  Fig.  86,  so  that  the  value  of  P  is  2F9  which  in  this 
case  is  the  same  as  the  resultant  pressure. 

The  formulas  here  deduced  are  entirely  independent  of  the 
form  of  the  surface,  and  of  the  angle  with  which  the  jet  enters 
upon  it.  It  is  clear,  however,  if,  as  in  the  planes  in  Fig.  86,  this 
angle  is  such  as  to  allow  shock  to  occur,  that  foam  and  changes 
in  cross-section  may  result  which  will  cause  energy  to  be  dissi- 
pated in  heat.  These  losses  will  diminish  the  velocity  v  and 
decrease  the  theoretic  dynamic  pressure.  These  effects  cannot 
be  formulated,  but  it  is  a  general  principle,  which  is  confirmed 


ART.  131.]  CURVED  PIPES  AND   CHANNELS.  3T3 

by  experiment,  that  they  may  be  largely  avoided  by  allowing 
the  jet  to  impinge  tangentially  upon  the  surface. 

In  all  the  foregoing  formulas  the  weight  W  which  im- 
pinges upon  the  surface  per  second  is  the  same  as  that  which 
issues  from  the  orifice  or  nozzle  that  supplies  the  stream,  and 
its  value  is 

W  =  wq  =  wav. 

To  find  W  it  is  hence  necessary  to  determine  the  discharge  q 
by  the  methods  explained  in  the  preceding  chapters,  or  to 
measure  a,  the  area  of  the  cross-section  of  the  stream,  and  to 
ascertain  by  some  method  the  mean  velocity  v. 

Prob.  156.  If  F  is  10  pounds,  OL  =  o°,  and  6  =  60°,  show 
that  the  pressure  parallel  to  the  direction  of  the  jet  is  5 
pounds,  that  the  pressure  normal  to  that  direction  is  8.66 
pounds,  and  that  the  resultant  dynamic  pressure  is  10  pounds. 


ARTICLE  131.  CURVED  PIPES  AND  CHANNELS. 

The  dynamic  pressures  discussed  in  the  preceding  article 
have  been  those  caused  by  jets,  or  isolated  streams,  of  water. 
There  is  now  to  be  considered  the  case  of  dynamic  pressures 
caused  by  streams  flowing  in  pipes,  conduits,  or  channels  of  any 
kind  ;  these  are  sometimes  called  limited  or  bounded  streams, 
the  boundary  being  the  surface  whose  cross-section  is  the  wetted 
perimeter.  When  such  a  stream  is  straight  and  of  uniform 
section,  and  all  its  filaments  move  with  the  same  velocity  v,  the 
impulse,  or  the  pressure  which  it  can  produce,  is  the  quantity 
F  given  by  the  general  expression  in  Art.  128;  under  these 
conditions  it  exerts  no  dynamic  pressure,  but  if  a  body  be  im- 
mersed and  held  stationary,  pressure  is  produced  upon  it.  If 
its  direction  changes  in  an  elbow  or  bend,  pressure  upon  the 


3 14        DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  XI. 

bounding  surface   is    produced ;    if   its    cross-section  increases 
or  decreases,  pressure  is  developed  or  absorbed. 

The  resultant  dynamic  pressure  P'  upon  a  curved  pipe 
which  runs  full  of  water  with  the  uniform  velocity  v  depends 
upon  the  angle  6  between  the  initial  and  final  directions,  and 


FIG.  89. 


must  be  the  same  as  that  produced  upon  a  surface  by  an  im- 
pinging jet  which  passes  over  it  without  change  in  velocity. 
The  formula  of  Art.  130  then  directly  applies,  and 


Pr  =  2  sin    6  .  F  =  2  sin 


W-  . 


If  0  =  o°,  there  is  no  bend,  and  Pr  =  o  ;  if  8  =  180°,  the  direc- 
tion of  flow  is  reversed,  and  P'  =  2F.  If  the  direction  is  twice 
reversed,  as  at  C  in  Fig.  89,  the  value  of  6  is  360°,  and  the  re- 
sultant is  the  initial  impulse  F  minus  the  final  reaction  F,  or 
simply  zero  ;  in  this  case,  however,  there  may  be  a  couple 
which  tends  to  twist  the  pipe,  unless  the  impulse  and  reaction 
lie  in  the  same  line. 

The  total  dynamic  pressure  exerted  upon  the  curved  pipe 
may  be  found  by  taking  the  sum  of  all  the  elementary  radial 
pressures.  For  this  purpose  let  the  pipe  at  A  in  Fig.  89  have 
the  length  dl  and  let  6  be  nearly  o°,  so  that  its  value  is  the  ele- 
mentary angle  66.  Then  in  the  above  formula  P'  becomes  the 
elementary  radial  pressure  6Pl  ,  and 

dP  =  2  sin    66  .  F  =  F60. 


ART.  131.] 


CURVED  PIPES  AND    CHANNELS. 


315 


Now  for  a  circular  curve  whose  radius  is  R,  the  value  of  SI  is 
R66 ;  and  accordingly  the  elementary  radial  pressure  for  that 
case  is  expressed  by  the  dffferential  equation 


The  total  radial  pressure  Pl  upon  a  circular  curve  whose 
length  is  /  is  the  integral  of  this  equation  between  the  limits  o 
and  Pl  for  the  first  member  and  o  and  /  for  the  second,  or 

p__M=  i^_ 

1  ~~  R  a  R  2g ' 

This  dynamic  pressure  does  no  work  and  offers  no  direct  re- 
sistance in  the  direction  of  the  flow. ;  but  in  being  transmitted 
through  the  water  to  the  outer  side  of  the  pipe  it  causes  cross- 
currents which  consume  energy.  This  expression  for  radial 
pressure  is  the  same  as  that  given  by  the  theory  of  centrifugal 
force.  It  is  not  strictly  exact  unless  all  the  filaments  have  the 
same  velocity  v,  which  in '  a  curved  pipe  is  probably  never 
the  case. 

The  same  reasoning  applies  approximately  to  the  curves 
of  conduits,  canals,  and  rivers.  In  any  length  /  there  exists 
a  radial  dynamic  pressure  Plt  acting  toward  the  outer  bank 
and  causing  currents  in  that 
direction,  which,  in  connec- 
tion with  the  greater  ve- 
locity that  naturally  there 
exists,  tends  to  deepen  the 
channel  on  that  side.  This 
may  help  to  explain  the  rea- 
son why  rivers  run  in  winding 
courses.  At  first  the  curve  FIG.  90. 

may  be  slight,  but  the  radial  flow  due  to  the  dynamic  pres- 


/|  ^      jj 

"f~  j----i-_ 


3l6        DYNAMIC  PRESSURE   OF  FLOWING    WATER.    [CHAP.  XL 

sure  causes  the  outer  bank  to  scour  away ;  this  increases  the 
velocity  v9  and  decreases  vl  (Fig.  90),  and  this  in  turn  hastens 
the  scour  on  the  outer  and  allows  material  to  be  deposited  on 
the  inner  side.  Thus  the  process  continues  until  a  state  of 
permanency  is  reached,  and  then  the  existing  forces  tend  to 
maintain  the  curve.  The  cross-currents  which  the  radial  pres- 
sure produces  have  been  actually  seen  in  experiments  devised 
by  THOMSON,*  and  it  is  found  that  they  move  in  the  manner 
shown  in  Fig.  90,  the  motion  toward  the  outer  bank  being  in 
the  upper  part  of  the  section,  while  along  the  wetted  perimeter 
they  flow  toward  the  inner  bank. 

The  elevation  of  the  water  on  the  outer  bank  of  a  bend 
is  higher  than  on  the  inner.  This  is  only  a  practical  conse- 
quence of  the  radial  dynamic  pressure,  as  in  straight  streams 
it  is  also  seen  that  the  water  surface  is  curved,  the  highest 
elevation  being  where  the  velocity  is  greatest.  In  this  case 
cross-currents  are  also  observed  which  move  near  the  surface 
toward  the  centre  of  the  stream,  and  near  the  bottom  toward 
the  banks,  their  motion  being  due  to  the  disturbance  of  the 
static  pressure  consequent  upon  the  varying  water  level. 

Prob.  157.  Why  is  it  that  streams  of  slight  slope  have  the 
most  winding  courses? 


ARTICLE  132.  IMMERSED  BODIES. 

When  a  body  is  immersed  in  a  flowing  stream,  or  when  it 
is  moved  in  still  water,  so  that  filaments  are  caused  to  change 
their  direction,  a  dynamic  pressure  is  exerted  or  overcome. 
The  theoretic  determination  of  the  intensity  of  this  pressure  is 
difficult,  if  not  impossible,  and  will  not  be  here  attempted; 

*  Proceedings  Royal  Society,  1877,  P-  35°- 


ART.  132.] 


IMMERSED  BODIES. 


317 


in  fact,  experiment  alone  can  furnish  reliable  conclusions.  It 
is,  however,  to 
be  inferred  from 
what  has  .  pre- 
ceded, that  the 
dynamic  pressure  FIG.  9i. 

in  the  direction  of  the  motion  is  proportional  to  the  force  of 
impulse  of  a  stream  whose  cross-section  is  the  same  as  that  of 
the  body,  or 

P  =  m  .  wa  — , 

in  which  m  is  a  number  depending  upon  the  length  and  shape 
of  the  immersed  portion,  and  whose  value  is  2  for  a  jet  imping- 
ing normally  upon  a  plane. 

Experiments  made  upon  small  plates  held  normally  to  the 
direction  of  the  flow  show  that  the  value  of  m  lies  between 
1.25  and  1.75,  the  best  determinations  being  near  1.4  and  1.5. 
It  is  to  be  expected  that  the  dynamic  pressure  on  a  plate  in  a 
stream  would  be  less  than  that  due  to  the  impulse  of  a  jet  of 
the  same  cross-section,  as  the  filaments  of  water  near  the  outer 
edges  are  crowded  sideways,  and  hence  do  not  impinge  with 
full  normal  effect,  and  the  above  results  confirm  this  supposi- 
tion. The  few  experiments  on  record  were  made  with  small 
plates,  mostly  less  than  2  square  feet  area,  and  they  seem  to  in- 
dicate that  in  is  greater  for  large  surfaces  than  for  small  ones. 

The  determination  of  the  dynamic  pressure  upon  the  end  of 
a  cylinder,  as  at  B  in  Fig.  91,  is  difficult  because  of  the  resist- 
ing friction  of  the  sides ;  but  it  is  well  ascertained  to  be  less 
than  that  upon  a  plane  of  the  same  area,  and  within  certain 
limits  to  decrease  with  the  length.  For  a  conical  or  wedge- 
shaped  body  the  dynamic  pressure  is  less  than  that  upon  the 
cylinder,  and  it  is  found  that  its  intensity  is  much  modified 
by  the  shape  of  the  rear  surface. 


3l8         DYNAMIC  PRESSURE    OF  FLOWING    WATER.    [CHAP.  XL 

When  a  body  is  so  formed  as  to  gradually  deflect  the  fila- 
ments of  water  in  front,  and  to  allow  them  to  gradually  close  in 
again  upon  the  rear,  the  impulse  of  the  front  filaments  upon  the 
body  is  balanced  by  the  reaction  of  those  in  the  rear,  so  that  the 
resultant  dynamic  pressure  is  zero.  The  forms  of  boats  and 
ships  should  be  made  so  as  to  secure  this  result,  and  then  the 
propelling  force  has  only  to  overcome  the  frictional  resistance 
of  the  surface  upon  the  water. 

The  dynamic  pressure  produced  by  the  impulse  of  ocean 
waves  striking  upon  piers  or  lighthouses  is  often  very  great. 
The  experiments  of  STEVENSON*  on  Skerryvore  Island,  where 
the  waves  probably  acted  with  greater  force  than  usual,  showed 
that  during  the  summer  months  the  mean  dynamic  pressure 
per  square  foot  was  about  600  pounds,  and  during  the  winter 
months  about  2100  pounds,  the  maximum  observed  value  be- 
ing 6100  pounds.  At  the  Bell  Rock  lighthouse  the  greatest 
value  observed  was  about  3000  pounds  per  square  foot.  The 
observations  were  made  by  allowing  the  waves  to  impinge 
upon  a  circular  plate  about  6  inches  in  diameter,  and  the  pres- 
sure produced  was  registered  by  the  compression  of  a  spring. 

Prob.  158.  Compute  the  probable  dynamic  pressure  upon  a 
surface  one  foot  square  when  immersed  in  a  current  whose 
velocity  is  8  feet  per  second,  the  direction  of  the  current  being 
normal  to  the  surface. 

ARTICLE  133.  MOVING  VANES. 

A  vane  is  a  plane  or  curved  surface  which  moves  in  a  given 
direction  under  the  dynamic  pressure  exerted  by  an  impinging 
jet  or  stream.  The  direction  of  the  motion  of  the  vane  de- 
pends upon  the  conditions  of  its  construction  ;  for-  example, 
the  vanes  of  a  water  wheel  can  only  move  in  a  circumference 
around  its  axis.  The  simplest  case  for  consideration,  however, 

*  RANKINE'S  Civil  Engineering,  p.  756. 


ART.  133.]  MOVING    VANES.  319 

is  that  where  the  motion  is  in  a  straight  line,  and  this  alone 
will  be  considered  in  this  article.  The  plane  of  the  stream  and 
vane  is  to  be  taken  as  horizontal,  so  that  no  direct  action  of 
gravity  can  influence  the  action  of  the  jet. 

Let  a  jet  with  the  velocity  v  impinge  upon  a  vane  which 
moves  in  the  same  direction  with  the  velocity  u,  and  let  the 
velocity  of  the  jet  relative  to  the  surface 
at  the  point  of  exit  make  an  angle  ft  with 
the  reverse  direction  of  u,  as  shown  in 
Fig.  92.     The  velocity  of  the  stream  rela- 
tive to  the  surface  is  v  —  u,  and  the  dy- 
namic pressure  is  the  same  as  if  the  sur- 
face were  at  rest,  and  the  stream  moving 

with  the  absolute  velocity  v  —  u.  Hence  formula  (84)  directly 
applies,  replacing  v  by  v  —  u,  and  0  by  —  /?,  and 


/>=(!+  COS  /S)  fF  —  • 


*  '     i  ti&  6  i  ^  -V-ta 

In  this  formula  W  is  not  the  weight  of  the  water  which  issues  ' 
from  the  nozzle,  but  that  which  strikes  and  leaves  the  vane,  or 
W  =  ^va(v  —  u)  ;   for  under  the  condition  here  supposed  the 
vane  moves  continually  away  from  the  nozzle,  and  hence  does 
not  receive  all  the  water  which  it  delivers. 

Another  method  of  deducing  the  last  equation  is  as  follows  : 
At  the  point  of  exit  let  lines  be  drawn  representing  the  veloci- 
ties v  —  u  and  it  ;  then  completing  the  parallelogram,  the  line 
vl  is  the  absolute  velocity  of  the  departing  jet  (Art.  33).  Let 
6  be  the  angle  which  vl  makes  with  the  direction  of  u,  and  ft 
as  before  the  angle  between  v  —  u  and  the  reverse  direction  of 
u.  Then  the  dynamic  pressure  is  that  due  to  the  absolute  im- 
pulse of  the  entering  and  departing  streams  ;  the  former  of 

W  W 

these  has  the  value  —  v  and  the  latter  the  value  —  v  cos  0. 


320  D  YNAMIC  PRESSURE  OF  FLO  WING  WA  TER.  [CHAP.  XL 
Hence  it  is  expressed  by  the  formula 

W 
P  =  —  0  —  ^  cos  0). 

o 

But  from  the  triangle  between  v1  and  u 

vl  cos  6  —  u  —  (v  —  u)  cos  /?. 
Inserting  this,  the  value  of  the  dynamic  pressure  is 

W 
P  =  —(v-u)(t+cos/3),          L.W'lM. 

o 

which  is  the  same  as  that  found  before.  If  §  =  180°  there  is 
no  pressure,  and  if  fi  =  o°  the  stream  is  completely  reversed, 
and  P  attains  its  maximum  value.  The  dynamic  pressure  may 
be  exerted  with  different  intensities  upon  different  parts  of  the 
vane,  but  its  total  value  in  the  direction  of  the  motion  is  that 
given  by  the  formula. 

Usually  the  direction  of  the  motion  is  not  the  same  as  that 
of  the  jet.  This  case  is  shown  in  Fig.  93,  where  the  arrow 
marked  F  designates  the  direction  of  the  impinging  jet,  and 
that  marked  P  the  direction  of  the  motion  of  the  vane,  a  be- 

ing  the  angle  between  them. 
The  jet  having  the  velocity  v 
impinges  upon  the  vane  at  A, 
and  in  passing  over  it  exerts 
a  dynamic  pressure  P  which 
causes  it  to  move  with  the 
velocity  u.  At  A  let  lines  be 
drawn  representing  the  inten- 
FIG.  93.  sities  and  directions  of  v  and 

«,  and  let  the  parallelogram  of  velocities  be  formed  as  shown ; 
the  line  V  then  represents  the  velocity  of  the  water  relative  to 
the  vane  at  A.  The  stream  passes  over  the  surface  and  leaves 
it  at  B  with  the  same  relative  velocity  V,  if  not  retarded  by 


ART.  133.]  MOVING    VANES.  321 

friction  or  foam.  At  B  let  lines  be  drawn  to  represent  u  and 
F,  and  let  /3  be  the  angle  which  V  makes  with  the  reverse 
direction  of  u  ;  let  the  parallelogram  be  completed,  giving  v^ 
for  the  absolute  velocity  of  the  departing  water,  and  let  6  be 
the  angle  which  it  makes  with  u.  The  total  pressure  in  the 
direction  of  the  motion  is  now  to  be  regarded  as  that  caused 
by  the  components  in  that  direction  of  the  initial  and  the  final 
impulse  of  the  water.  The  impulse  of  the  stream  before  strik- 

W 
ing  the  vane  is  —  v,  and  its  component  in  the  direction  of  the 

o 

W 

motion  is  —  v  cos  a.     That  of  the  stream  as  it  leaves  the  vane 
g 

W 
is  —  vlf  and  its  component  upon  the  direction  of  the  motion  is 

o 

W 

—  z\  cos  6.     The  difference  of  these  components  is  the  total 

pressure  in  the  given  direction,  or 

W 

P  —  —  (v  cos  a  —  v^  cos  0)  .....     (85) 

o 

This  is  a  general  formula  for  the  pressure  in  any  given  direc- 
tion upon  a  vane  moving  in  a  straight  line.  If  the  surface  be 
at  rest  v^  equals  v,  and  it  agrees  with  the  result  deduced  in 
Art.  130. 

If  it  be  required  to  find  the  numerical  value  of  P,  the  given 
data  are  the  velocities  v  and  «,  and  the  angles  a  and  /?.  The 
term  vl  cos  0  is  hence  to  be  expressed  in  terms  of  these  quanti- 
ties. From  the  triangle  at  B  between  vl  and  u 

vl  cos  0  =  n  —  V  cos  ft. 
Substituting  this,  the-  formula  becomes 
W 


.     .     .     (85)' 

<*> 


322         DYNAMIC  PRESSURE    OF  FLOWING   WATER.    [CHAP.  XL 

which  is  often  a  more  convenient  form  for  discussion.  The 
value  of  V  is  found  from  the  triangle  at  A  between  u  and  v,  thus  : 

V^  =  u*  -\-  v*  —  2uv  cos  a  ; 

and  hence  the  dynamic  pressure  P  is  fully  determined  in  terms 
of  the  given  data. 

In  order  that  the  stream  may  enter  tangentially  upon  the 
vane,  and  thus  prevent  foam,  the  curve  of  the  vane  at  A  should 
be  tangent  to  the  direction  of  F.  This  direction  can  be  found 
by  expressing  the  angle  0  in  terms  of  the  given  angle  a.  Thus 
from  the  relation  between  the  sides  and  angles  of  the  triangle 
included  between  u,  v,  and  V  there  is  found 

sin  (0  —a)  __  u 
sin  0        ~  v  ' 

which  reduces  to  the  form 

u 

cot  0  =  cot  a : 

v  sin  a 

from  which  0  can  be  computed  when  u,  v,  and  a  are  given.  If 
the  angle  made  by  the  vane  with  the  direction  of  the  motion 
be  greater  or  less  than  0  some  loss  due  to  impact  will  result. 

Prob.  159.  What  does  <vl  represent  in  the  parallelogram 
drawn  at  B  in  Fig.  93  ?  Express  its  value  in  terms  of  /?,  u, 
andF, 


ARTICLE  134.    WORK  DERIVED  FROM  MOVING  VANES. 

The  work  imparted  to  a  moving  vane  by  the  energy  of  the 
impinging  water  is  equal  to  the  product  of  the  dynamic  pres- 
sure P}  which  is  exerted  in  the  direction  of  the  motion  and  the 
space  through  which  it  moves.  If  u  be  the  space  described  in 
one  second,  or  the  velocity  of  the  vane,  the  work  per  second  is 

k-Pu. 


ART.  134.]     WORK  DERIVED  FROM  MOVING    VANES.  323 

This  expression  is  now  to  be  discussed  in  order  to  determine 
the  value  of  u  which  makes  k  a  maximum. 


When  the  vane  moves  in  a  straight  line  in  the  same  direc- 
tion as  the  impinging  jet  and  the  water  enters  it  tangentially, 
as  shown  in  Fig.  87,  the  work  imparted  is  found  by  inserting  for 
P  its  value  from  (84),  whence 


4      k^  (I  +  cos  P)  W  =  (I  +  cos  ftwa         -         . 

o  o 

The  value  of  u  which  renders  k  a  maximum  is  obtained  by 
equating  to  zero  the  derivative  of  k  with  respect  to  u,  or 

fik  wet  • 

—  =  (i  +  cos  ft)  —  0'  -  4vu  +  3<)  =  o, 

C> 

from  which  the  value  of  u  is 


u  =  $v; 
and  inserting  this,  the  maximum  work  is  found  to  be 

wa  if 
k  —  8(1  +cos/3)  --  . 

^  27  2g 

The  theoretic  energy  of  the  impinging  jet  is 

9  3 


and  accordingly  the  efficiency  of  the  vane  is  (Art.  31) 

k         8  , 
e  =  -g  =  --(i  +  cos  ft). 

If  /?  —  1  80°,  the  jet  glides  along  the  vane  without  producing 
work  and  e  =  o  ;  if  ft  =  90°,  the  water  departs  from  the  vane 
normal  to  its  original  direction  and  e  =  -fa  ;  if  /?  =  o,  the  direc- 
tion of  the  stream  is  reversed  and  e  =  -|-f  . 

It  appears  from  the  above  that  the  greatest  efficiency  which 
can  be  obtained  by  a  vane  moving  in  a  straight  line  under  the 
impulse  of  a  jet  of  water  is  1$  ;  that  is,  the  effective  work  is 
only  about  59  per  cent  of  the  theoretic  energy  attainable. 


324        DYNAMIC  PRESSURE   OF  FLOWING  WATER.    [CHAP.  XI. 

This  is  due  to  two  causes  :  first,  the  quantity  of  water  which 
reaches  and  leaves  the  vane  per  second  is  less  than  that  fur- 
nished by  the  nozzle  or  mouthpiece  from  which  the  water 
issues  ;  and  secondly,  the  water  leaving  the  wane  still  has  an  ab- 
solute velocity  of  ^v.  A  vane  moving  in  a  straight  line  is 
therefore  a  poor  arrangement  for  utilizing  energy,  and  it  will 
also  be  seen  upon  reflection  that  it  would  be  impossible  to  con- 
struct a  motor  in  which  a  vane  would  move  continually  in  the 
same  direction  away  from  a  fixed  nozzle.  The  above  discussion 
therefore  gives  but  a  rude  approximation  to  the  results  ob- 
tainable under  practical  conditions.  It  shows  truly,  however, 
that  the  efficiency  of  a  jet  which  is  deflected  normally  from  its 
path  is  but  one  half  of  that  obtainable  when  a  complete  reversal 
of  direction  is  made. 

r  Water  wheels  which  act  under  the  impulse  of  a  jet  consist 
of  a  series  of  vanes  arranged  around  a  circumference  which  by 
the  motion  are  brought  in  succession  before  the  jet.  In  this 
case  the  quantity  of  water  which  leaves  the  wheel  per  second 
is  the  same  as  that  which  enters  it,  so  that  W  does  not  depend 
on  the  velocity  of  the  vanes,  as  in  the  preceding  case,  but  is  a 
constant  whose  value  is  wq,  where  q  is  the  quantity  furnished 
per  second.  An  approximate  estimate  of  the  efficiency  of  a 
series  of  such  vanes  can  be  made  by  considering  a  single  vane 
and  taking  W  as  a  constant.  The  water  is  supposed  to  im- 
pinge tangentially,  and  the  vane  to  move  in  the  same  direction 
as  the  jet  (Fig.  92).-  Then  the  work  imparted  per  second  is 


This  becomes  zero  when  u  =  o  or  when  u  =  v,  and  it  is  a 
maximum  when  u  =  %v,  or  when  the  vane  moves  with  one- 
half  the  velocity  of  the  jet.  Inserting  this  value  of  u, 

k  =  (i  +  cos  /3)W     -; 


ART.  134.]      WORK  DERIVED  FROM  MOVING    VANES.  $2$ 

and  dividing  this  by  the  theoretic  energy  W —  ,  the  efficiency  is 
e  =  i(i  +  cos  ft). 

When  ft  =  1 80°,  the  jet  merely  glides  along  the  surface  with- 
out performing  work  and  e  —  o ;  when  ft  =  90°,  the  jet  is 
deflected  normally  to  the  direction  of  the  motion  and  e  =  $ ; 
when  ft  =  o°,  a  complete  reversal  of  direction  is  obtained  and 
the  efficiency  attains  its  maximum  value  e  =  :. 

These  conclusions  apply  approximately  to  the  vanes  of  a 
water-wheel  which  are  so  shaped  that  the  water  enters  upon 
them  tangentially  in  the  direction  of  the  motion.  If  the  vanes 
are  plane  radial  surfaces,  as  in  simply  paddle-wheels,  the  water 
passes  away  normally  to  the  circumference  and  the  highest 
obtainable  efficiency  is  about  50  per  cent.  If  the  vanes  are 
curved  backward  the  efficiency  becomes  greater,  and,  neglect- 
ing losses  in  impact  and  friction,  it  might  be  made  nearly  unity, 
and  the  entire  energy  of  the  stream  be  realized,  if  the  water 
could  both  enter  and  leave  the  vanes  in  a  direction  tangent  to 
the  circumference.  The  investigation  shows  that  this  is  due 
to  the  fact  that  the  water  leaves  the  vanes  without  velocity ; 
for,  as  the  advantageous  velocity  of  the  vane  is  %v,  the  water 
upon  its  surface  has  the  relative  velocity  v  —  \v  =  \v ;  then,  if 
ft  =  o,  as  it  leaves  the  vane  its  absolute  velocity  is  \v  —  \v  =  o. 
If  the  velocity  of  the  vanes  is  less  or  greater  than  half  the 
velocity  of  the  jet,  the  efficiency  is  lessened,  although  slight 
variations  from  the  advantageous  velocity  do  not  practically 
influence  the  value  of  e. 

Prob.  1 60.  A  nozzle  0.125  feet  in  diameter,  whose  coeffi- 
cient of  discharge  is  0.95,  delivers  water  under  a  head  of  82 
feet  against  a  series  of  small  vanes  on  a  circumference  whose 
diameter  is  18.5  feet.  Find  the  most  advantageous  velocity  of 
revolution. 


326        DYNAMIC  PRESSURE    OF  FLOWING   WATER.     [CHAP.  XI. 

ARTICLE  135.  REVOLVING  VANES. 

When  vanes  are  attached  to  an  axis  around  which  they 
move,  as  is  the  case  in  water  wheels,  the  dynamic  pressure 
which  is  effective  in  causing  the  motion  is  that  tangential  to 
the  circumferences  of  revolution  ;  or  at  any  given  point  this 
effective  pressure  is  normal  to  a  radius  drawn  from  the  point  to 
the  axis.  In  Fig.  94  are  shown  two  cases  of  a  rotating  vane  ; 
in  the  first  the  water  passes  outward  or  away  from  the  axis, 
and  in  the  second  it  passes  inward  or  toward  the  axis.  The 
reasoning,  however,  is  general  and  will  apply  to  both  cases. 
At  Ay  where  the  jet  enters  upon  the  vane,  let  v  be  its  absolute 
velocity,  V  its  velocity  relative  to  the  vane,  and  u  the  velocity 
of  the  point  A  ;  draw  u  normal  to  the  radius  r  and  construct 
the  parallelogram  of  velocities  as  shown,  a  being  the  angle  be- 
tween the  directions  of  u  and  v,  and  0  that  between  u  and  V. 


FIG.  94. 

At  B,  where  the  water  leaves  the  vane,  let  u1  be  the  velocity  of 
that  point  normal  to  the  radius  rlt  and  Vl  the  velocity  of  the 
water  relative  to  the  vane ;  then  constructing  the  parallelogram, 
the  resultant,  of  «x  and  Vl  is  vl9  the  absolute  velocity  of  the 
departing  water.  Let  /?  be  the  angle  between  V^  and  the 
reverse  direction  of  ul ,  and  6  be  the  angle  between  the  direc- 
tions of  v,  and  u,  . 


ART.  135.]  REVOLVING    VANES.  327 

The  total  dynamic  pressure  exerted  in  the  direction  of 
the  motion  will  depend  upon  the  values  of  the  impulse  in  the 
entering  and  departing  streams.  .  The  absolute  impulse  of  the 

W 
water  before  entering  is  — v,  and  that  of  the  water  after  leav- 

<5 

W 

ing  is  — TV     Let  the  components  of  these  in  the  direction  of 

o 

the  motion  be  designated  by  P  and  Pl  ;  then, 

W  W 

P  =  - — v  cos  a,  P  =  — v   cos  0. 

g  <  g 

These,  however,  cannot  be  subtracted  to  give  the  resultant 
dynamic  pressure,  as  was  done  in  the  case  of  motion  in  a 
straight  line,  because  their  directions  are  not  parallel,  and  the 
velocities  of  their  points  of  application  are  not  equal.  The 
resultant  dynamic  pressure  is  not  important  in  cases  of  this 
kind,  but  the  above  values  will  prove  very  useful  in  the  next 
article  in  investigating  the  work  that  can  be  performed  by  the 
vane. 

The  given  data  for  a  revolving  vane  are  the  angles  a  and  /?, 
and  the  velocities  v,  u,  and  u^  To  find  the  auxiliary  angle  0  the 
triangle  at  B  between  #x  and  Vl  gives 

Vl  COS   B  =   Ul  —    Fj  COS  /?. 

When  the  motion  is  in  a  straight  line  the  relative  velocities  V 
and  Fj  are  equal,  if  the  friction  is  so  slight  that  it  can  be 
neglected  ;  for  a  revolving  vane,  however,  they  are  unequal,  and 
the  relation  between  them  will  be  deduced  in  the  next  article. 

If  n  be  the  number  of  revolutions  around  the  axis  in  one 
second,  the  velocities  u  and  ul  are 

0 

u  =  27trn,  u^  =  27trln, 

and  accordingly  the  relation  obtains, 

u        r 
u'~  r' 


328         DYNAMIC  PRESSURE    OF  FLOWING   WATER.    [CHAP.  XL 

or  the  velocities  of  the  points  of  entrance  and  exit  are  directly 
proportional  to  their  distances  from  the  axis.  If  r  and  rl  are 
both  infinity,  //  equals  «,  and  the  case  is  that  of  motion  in  a 
straight  line  as  discussed  in  Art.  133. 

Prob.  161.  If  a  point  14  inches  from  the  axis  moves  with  a 
uniform  velocity  of  62  feet  per  second,  how  many  revolutions 
does  it  make  per  minute  ? 

ART.  136.  WORK  DERIVED  FROM  REVOLVING  VANES. 

The  investigation  in  Art.  134  on  the  work  and  efficiency  of  a 
revolving  vane  supposes  that  all  its  points  move  with  the  same 
velocity,  and  that  the  water  enters  upon  it  in  the  same  direc- 
tion as  that  of  its  motion,  or  that  a  =  o.  This  cannot  in 
general  be  the  case  in  water  motors,  as  then  the  jet  would  be 
tangential  to  the  circumference  and  no  water  could  enter.  To 

o 

consider  the  subject  further  the  reasoning  of  the  last  article 
will  be  continued,  and,  using  the  same  notation,  it  will  be  plain 
that  the  work  may  be  regarded  as  that  due  to  the  impulse  of 
the  entering  stream  in  the  direction  of  the  motion  around  the 
axis  minus  that  due  to  the  impulse  of  the  departing  stream  in 
the  same  direction,  or 

k  =  Pu  —  P,u,. 

Here  P  and  P^  are  the  pressures  due  to  the  impulse  at  A  and 
B  (Fig.  94),  and  inserting  their  values  as  found, 

W 
k  =  — (uv  cos  a  —  ulvl  cos  B) (86) 

o 

This  is  a  general  formula  applicable  to  the  work  of  all  wheels 
of  outward  or  inward  flow,  and  it  is  seen  that  the  useful  work 
k  consists  of  two  parts,  one*  due  to  the  entering  and  the  other 
to  the  departing  stream. 

Another  general  expression  for  the  work  of  a  series  of  vanes 
may  be  established  as  follows :  Let  v  and  vl  be  the  absolute 


ART.  136.]    WORK  DERIVED  FROM  REVOLVING    VANES.          329 

velocities  of  the  entering  and  departing  water ;  then  the  theo- 

v* 
retic   energy  is    W — ,  and  there  is  carried  away  the   energy 

V  *  ^ 

W-—.     The  difference  of  these  is  the  work  imparted  to  the 

& 
wheel,  neglecting  losses  of  energy  in  friction  and  impact,  or 

W 
k  =  —V-v?) (87) 

This  is  a  formula  of  equal  generality  with  ,the  preceding,  and 
like  it  is  applicable  to  all  cases  of  the  conversion  of  energy  into 
work  by  means  of  impulse  or  reaction.  In  both  formulas,  how- 
ever, the  plane  of  the  vane  is  supposed  to  be  horizontal,  so 
that  no  fall  occurs  between  the  points  of  entrance  and  exit. 

A  useful  relation  between  the  relative  velocities  V  and  V^ 
can  be  deduced  by  equating  the  values  of  k  given  by  the  pre- 
ceding formulas  ;  thus : 

uv  cos  a  —  u^  cos  0  =  -J-(Va  —  vf). 
Now  from  the  triangle  at  A  between  u  and  v 
v*  =  V*  —  if  -\-  2uv  cos  a, 
and  from  the  triangle  at  B  between  ul  and  v^ 
v?  =  V?  -  u?  +  2u,v,  cos  6. 
Inserting  these  values  of  v*  and  v*  the  relation  reduces  to 

V?  -  u?  =  V  -  u* (88) 

This  is  the  formula  by  which  the  relative  velocity  Vl  of  the 
issuing  water  is  to  be  computed  when  V  is  given.  It  shows 
when  «,  =  u  that  F,  =  F,  as  is  the  case  in  Fig.  93,  where  the 
motion  is  in  a  straight  line.  If,  however,  ut  be  greater  than  uy 
as  in  the  outward-flow  vane  of  the  first  diagram  of  Fig.  94, 
then  V^  is  greater  than  V;  if  ut  is  less  than  u,  as  in  an  inward- 
flow  vane,  then  Vl  is  less  than  V. 

The  above  principles  will  now  be  applied  to  the  simple  case 


33°        DYNAMIC  PRESSURE   OF  FLOWING  WATER.     [CHAP.  XI. 

of  a  vane  impinged  upon  tangentially  by  a  jet  which  passes  off 

off  in  a  radial  direction. 
The  two  diagrams  in 
Fig.  95  show  the  out- 
ward-flow and  the  in- 
ward-flow vane,  and  the 
reasoning  will  be  gen- 


eral, and  apply  to  both. 

*  "^       As  the  velocity  v  of  the 

FlG-95-  jet  has  the  same  direc- 

tion as  the  velocity  of  the  vane,  the  relative  velocity  V  at  the 
point  of  entrance  is  v  —  u.  The  work  imparted  to  the  vane 
by  the  jet  is,  from  (87), 

k  =  —W-v*\ 
~  ijg*  *»> 

From  the  parallelogram  drawn  at  the  point  of  exit, 

»,*  =  v?  + «;. 

But  from  the  relation  established  in  (88), 

whence  v*  is  found  to  be 

v*  —  v"  —  2uv  +  2u*. 
Hence  the  work  imparted  to  the  vane  is 

_  W  __  W 

~  ~g  U^~~~g 

This  becomes  zero  when  u  =  o,  or  when  u  =  — 5  v ;  and  it  is  a 

maximum  when 

-  Jk,  r- 
*\ 

This  advantageous  velocity  reduces  the  value  of  k  to 

W  —   a 


ART.  136.]    WORK  DERIVED  FROM  REVOLVING    VANES.          331 

If  r  =  o  the  work  k  vanishes,  as  it  should  do  ;  for  this  case  is 
that  of  an  outward-flow  vane,  where  the  water  has  only  a  radial 
motion.  If  r  =  r^  the  case  is  that  of  motion  in  a  straight  line, 
and  k  becomes  one-half  the  theoretic  energy  of  the  jet,  as  in 
Art.  134.  If  rl  —  o,  the  value  of  k  becomes  oo  ;  but  this  is  ab- 
surd, since  in  no  event  can  k  be  greater  than  W--  :  the  reason 

of  this  ridiculous  conclusion  lies  partly  in  the  fact  that  the  as- 
sumption r,  =  o  is  an  impossibility  for  an  inward-flow  vane, 
since  the  water  must  turn  aside  from  the  axis  before  reaching 
it,  and  partly  in  the  circumstance  that  the  advantageous  value 
of  u  was  deduced  by  supposing  jtl  finite.  Indeed,  if  r*  =  2r*, 
that  value  of  u  becomes  v,  and  plainly  &  cannot  exceed  v  if  the 
water  is  to  do  work  on  the  vane.  The  absurdity  therefore  may 
be  said  to  be  caused  by  the  fact  that  the  algebraic  maximum 
Jfor  this  case  lies  outside  the  limits  of  the  problem. 

Precisely  the  same  conclusions  may  be  drawn  from  the 
use  of  the  formula  (86)  instead  of  (87) ;  for  since  here  a  =  o, 
and  £\  cos  9  =  ult  it  reduces  to 

W 
k  =  —(uv-u?), 

which  is  the  same  as  before  found.  It  .appears  from  this  dis- 
cussion that  an  outward-flow  vane  under  the  conditions  of 
Fig.  95  cannot  utilize  more  than  one-half  the  energy  of  the 
jet,  but  that  an  inward-flow  vane  may  utilize  the  entire  energy. 
It  is  here  again  repeated,  that  the  effect  of  friction  and  foam 
have  been  neglected  in  the  investigation ;  these,  of  course,  tend 
to  make  the  velocity  Vl  less  than  its  theoretic  value,  and  thus 
consume  energy. 

Prob.  162.  In  the  first  diagram  of  Fig.  94  let  a  =  o,  0  =  o, 
fi  =  o,  and  r  —  o.  Prove  that  the  advantageous  velocity  is 
infinite. 


332        DYNAMIC  PRESSURE   OF  FLOWING  WATER.    [CHAP.  XL 

ARTICLE  137.  REVOLVING  TUBES. 

The  water  which  glides  over  a  vane  can  never  be  under 
static  pressure,  but  when  two  vanes  are  placed  near  together 
and  connected  so  as  to  form  a  closed  tube,  there  may  exist  in 
it  static  pressure  if  the  tube  is  filled.  This  is  the  condition  in 
turbine  wheels,  where  a  number  of  such  tubes,  or  buckets,  are 
placed  around  an  axis  and  water  is  forced  through  them  by 
the  static  pressure  of  a  head.  The  work  in  this  case  is  done 
by  the  dynamic  pressure  exactly  as  in  vanes,  but  the  existence 
of  the  static  pressure  renders  the  investigation  more  difficult. 

The  simplest  instance  of  a  revolving  tube  is  that  of  an  arm 
attached  to  a  vessel  rotating  about  a  vertical 
axis,  as  in  Fig.  96.  It  was  shown  in  Art.  29 
that  the  water  surface  in  this  case  assumes 
the  form  of  a  paraboloid,  and  if  no  discharge 
occurs  it  is  clear  that  the  static  pressures  at 
any  two  points  B  and  A  are  measured  by 
the  pressure-heads  Hl  and  H  reckoned  up- 
wards to  the  parabolic  curve,  and,  if  the  ve- 
locities of  those  points  are  u^  and  u,  that 


Now  suppose  an  orifice  to  be  opened  in  the 
end  of  the  tube  and  the  flow  to  occur  while 
at  the  same  time  the  revolution  is  continued.  The  velocities 
Fj  and  V  diminish  the  pressure-heads  so  that  the  piezometric 
line  is  no  longer  the  parabola  but  some  curve  represented  by  the 
lower  broken  line  in  the  figure.  Then  according  to  the  prin- 
ciple in  Art.  27,  that  pressure-head  plus  velocity-head  remains 
constant  if  no  loss  of  energy  occurs,  the  above  equation  be- 
comes 


(89) 


ART.  I37-]  REVOLVING    TUBES.  333 

in  which  H^  and  H  are  the  heads  due  to  the  actual  static  pres- 
sures. This  is  the  theorem  which  gives  the  relation  between 
pressure-head,  velocity-head,  and.  rotation-head  at  any  point 
of  a  revolving  tube  or  bucket.  If  the  tube  is  only  partly  full, 
1  so  that  the  flow  occurs  along  one  side,  like  that  of  a  stream 
upon  a  vane,  then  there  is  no  static  pressure,  Hl  =  o,  H  =  ol , 
and  the  formula  becomes  the  same  as  (88),  which  was  otherwise 
deduced  in  the  last  article. 

An  apparatus  like  Fig.  96,  but  having  a  number  of  arms 
from  which  the  flow  issues,  is  called  a  reaction  wheel,  since  the 
dynamic  pressure  which  causes  the  revolution  is  wholly  due  to 
the  reaction  of  the  issuing  water.  To  investigate  it,  the  gen- 
eral formula  (86)  may  be  used.  Making  u  =  o,  then  the  work 
done  upon  the  wheel  by  the  water  is 

W  W 

k  —  —  (—  u,v,  cos  0)  —  —  (z^Fj  cos  ft  —  u*). 

,  o  o 

But  since  there  is  no  static  pressure  at  the  point  B,  the  value 
of  Vl  is,  from  (89),  or  also  from  Art.  29, 


V,  =  V2gh  +  «,'. 

The  work  of  the  wheel  now  is 
W  , 


cos  /3  V2gh  +  u*  — 

o 


This  becomes  nothing  when  2il  =  o,  or  when  u*  =  2gh  cot8  yff, 
and  by  the  usual  method  it  is  found  that  it  becomes  a  maxi- 
mum when 


Inserting  this  advantageous  velocity,  the  corresponding  work  is 

k  =  Wh(\  -  sin  /?), 
and  therefore  the  efficiency  is 

e  =  i  —  sin  ft. 


334         DYNAMIC  PRESSURE   OF  FLOWING   WATER.     [CHAP.  XL 

When  ft  =  90°,  both  ul  and  e  become  o,  for  then  the  direction 
of  the  stream  is  normal  to  the  circumference  of  revolution  and 
no  reaction  can  occur.  When  ft  =  o  the  efficiency  becomes 
unity,  but  the  velocity  ul  becomes  infinity.  In  the  reaction 
wheel,  therefore,  high  efficiency  can  only  be  secured  by  making 
the  direction  of  the  issuing  water  directly  opposite  to  that 
of  the  revolution,  and  by  having  the  speed  very  great.  If 
ft  =  19°. 5  or  sin  ft  =  ^,  the  advantageous  velocity  ul  becomes 
\/2gh  and  e  becomes  0.67.  The  effect  of  friction  of  the  water 
on  the  sides  of  the  revolving  tube  is  not  here  considered,  but 
in  Art.  143,  where  the  reaction  wheel  is  to  be  further  discussed,, 
this  will  be  done. 

Prob.  163.  Compute  the  theoretic  efficiency  of  the  reaction 
wheel  when  6  =  180°,  ft  =  o°,  and  u,  = 


ART.  138.]  CONDITIONS  OF  HIGH  EFFICIENCY.  335 


CHAPTER  XII. 
HYDRAULIC   MOTORS. 

ARTICLE  138.  CONDITIONS  OF  HIGH  EFFICIENCY. 

There  are  three  ways  in  which  water  may  act  in  imparting 
its  energy  to  hydraulic  motors,  namely,  by  its  weight,  by  the 
dynamic  pressure  of  its  impulse  and  reaction,  and  by  its  static 
pressure.  To  the  first  class  belong  those  wheels  where  the 
water  descends  in  buckets,  to  the  second  impulse  wheels  and 
turbines,  and  to  the  third  those  in  which  pistons  are  moved  by 
static  pressure.  In  some  cases  both  weight  and  dynamic 
pressure  act  in  the  same  motor,  as  in  the  breast  wheel,  and  'to 
a  slight  extent  in  the  undershot.  The  following  pages  will  be 
devoted  to  a  discussion  of  some  of  the  most  important  motors, 
in  order  to  determine  the  conditions  which  render  them  most 
efficient. 

The  efficiency  e  of  a  motor  ought,  if  possible,  to  be  inde- 
pendent of  the  amount  of  water  used,  or  if  not,  it  should  be  the 
greatest  when  the  water  supply  is  low.  This  is  very  difficult  to 
attain.  It  should  be  noted,  however,  that  it  is  not  the  mere 
variation  in  the  quantity  of  water  which  causes  the  efficiency 
to  vary,  but  it  is  the  losses  of  head  which  are  consequent 
thereon.  For  instance,  when  water  is  low,  gates  must  be  low- 
ered to  diminish  the  area  of  orifices,  and  this  produces  sudden 
changes  of  section  which  diminish  the  effective  head  h.  A 

o 

complete  theoretic  expression  for  the  efficiency  will  hence  not 
include  W,  the  weight  of  water  supplied  per  second,  but  it 
should,  if  possible,  include  the  losses  of  energy  or  head  which 
result  when  W  varies.  The  actual  efficiency  of  a  motor  can 


3?6  HYDRAULIC  MOTORS.  [CHAP.  XII. 

only  be  determined  by  tests  with  a  friction  brake  ;  the  theoretic 
efficiency,  as  deduced  from  formulas  like  those  of  the  last 
chapter,  will  as  a  rule  be  higher  than  the  actual,  because  it  is 
impossible  to  formulate  accurately  all  the  sources  of  loss. 
Nevertheless,  the  deduction  and  discussion  of  formulas  for 
theoretic  efficiency  is  very  important  for  the  correct  under- 
standing and  successful  construction  of  hydraulic  motors. 

A  general  theoretic  expression  for  the  efficiency  will  now 
be  deduced.     The  theoretic  energy  per  second  is 

K=  WA=  W~. 

2£ 

The  actual  work  per  second  equals  the  theoretic  energy  minus 
all  the  losses  of  energy.  These  losses  may  be  divided  into  two 
classes  :  first,  those  caused  by  the  transformation  of  energy  into 
heat  ;  and  second,  those  due  to  the  velocity  v^  with  which  the 
water  reaches  the  level  of  the  tail  race.  The  first  class  includes 
losses  in  friction,  losses  in  foam  and  eddies  consequent  upon 
sudden  changes  in  cross-section,  or  from  allowing  the  entering 
water  to  dash  improperly  against  surfaces  -  let  the  loss  of  work 
due  to  this  be  Wk  ',  in  which  h'  is  the  head  lost  by  these  causes. 
The  second  loss  is  due-  merely  to  the  fact  that  the  departing 

water  carries  away  the  energy  W  —  .     The  work  per  second 

o 

imparted  to  the  wheel  then  is 

k=w(h-k'  - 
\ 

and  dividing  this  by  the  theoretic  energy,  the  efficiency  is 


This  formula,  although  very  general,  must  be  the  basis  of  all 
discussions  on  the  theory  of  water-wheels  and  motors.  It 
shows  that  e  can  only  become  unity  when  h!  =  o  and  vl  =  o, 


ART.  139.]  OVERSHOT    WHEELS.  337 

whence  the  two  following  fundamental  requirements  must  be 
fulfilled  in  order  to  secure  high  efficiency : 

1.  The  water  must  enter  and  pass  through  the  wheel  with- 

out losing  energy  in  friction  and  foam. 

2.  The  water  must  reach  the  level  of  the  tail  race  without 

absolute  velocity. 

These  two  requirements  are  expressed  in  popular  language  by 
the  maxim,  well  known  among  engineers,  "  the  water  must 
enter  the  wheel  without  shock  and  leave  without  velocity." 
Here  the  word  shock  means  that  method  of  introducing  the 
water  which  produces  foam  and  eddies. 

The  friction  of  the  wheel  upon  its  bearings  is  included  in 
the  lost  work  when  the  power  and  efficiency  are  actually 
measured  as  described  in  Art.  124.  But  as  this  is  not  a 
hydraulic  loss,  it  should  not  be  included  in  the  lost  work  k' 
when  discussing  the  wheel  merely  as  a  user  of  water,  as  will  be 
done  in  this  chapter.  The  amount  lost  in  shaft  and  journal 
friction  in  good  constructions  may  be  estimated  at  2  or  3  per 
cent  of  the  theoretic  energy,  so  that  in  discussing  the  hydraulic 
losses  the  maximum  value  of  e  will  not  be  unity,  but  about 
0.98  or  0.97.  This  may  perhaps  be  rendered  slightly  smaller 
by  the  friction  of  the  wheel  upon  the  air  or  water  in  which  it 
moves,  and  which  will  here  not  be  regarded. 

Prob.  164.  A  wheel  using  70  cubic  feet  per  minute  under  a 
head  of  12.4  feet  has  an  efficiency  of  0.63.  What  is  its  effective 
horse-power  ? 

ARTICLE  139.  OVERSHOT  WHEELS. 

In  the  overshot  wheel  the  water  acts  largely  by  its  weight. 
Fig.  97  shows  an  end  view  or  vertical  section,  which  so  fully 
illustrates  its  action  that  no  detailed  explanation  is  necessary. 
The  total  fall  from  the  surface  of  the  water  in  the  head  race  or 
flume  to  the  surface  in  the  tail  race  is  called  h.  The  weight  of 


338 


HYDRAULIC  MOTORS. 


[CHAP   XI 1 


water  delivered  per  second  is  represented  by  W\  then  the 
theoretic  energy  of  the  fall  per  second  is  Wh.  It  is  required 
to  determine  the  conditions  which  will  render  the  work  of  the 
wheel  as  near  to  Wh  as  possible. 


The  total  fall  may  be  divided  into  three  parts — that  in  which 
the  water  is  filling  the  buckets,  that  in  which  the  water  is 
descending  in  the  filled  buckets,  and  that  which  remains  after 
the  buckets  are  emptied.  Let  the  first  of  these  parts  be  called 
h0 ,  and  the  last  hl .  In  falling  the  distance  hQ  the  water  acquires 
a  velocity  v0  which  is  approximately  equal  to  V2gh0 ,  and  then 
striking  the  buckets  this  is  reduced  to  u,  the  tangential  velocity 
of  the  wheel,  whereby  a  loss  of  energy  in  impact  occurs.  It 
then  descends  through  the  distance  h  —  h0  —  h^  acting  by  its 
weight  alone,  and  finally  dropping  out  of  the  buckets,  reaches 
the  level  of  the  tail  race  with  a  velocity  which  causes  a  second 
I  loss  of  energy.  Let  h'  be  the  head  lost  in  entering  the  buckets, 
and  let  vl  be  the  velocity  of  the  water  as  it  reaches  the  tail 
race.  Then  the  efficiency  of  the  wheel  is  given  by  the  general 

formula  (90),  or 

h'      v" 


ART.  139.]  OVERSHOT    WHEELS.  339 

and  to  apply  it,  the  values  of  ti  and  vl  are  to  be  found.     In  this 
equation  v  is  the  velocity  due  to  the  head  h,  or  v  =  V2gk. 

The  head  lost  when  a  stream  of  water  with  the  velocity  % 
is  enlarged  in  section  so  as  to  have  the  smaller  velocity  #,  is,  as 
proved  in  Art.  68, 


The  velocity  v^  with  which  the  water  reaches  the  tail  race  de- 
pends upon  the  velocity  u  and  the  height  hl  .     Its  energy  as  it 

leaves  the  buckets  is  W  —  ,  and  that  required  in  the  fall  h^  is 

o 


4 ;  the  sum  of  these  must  be  equal  to  the  resultant  energy, 
W  -  — ,  whence  the  value  of  vl  is 


v,  =  Vu*  +  2gh,  . 

Inserting  these  values  of  hf  and  v^  in  the  formula  for  e,  and 
placing  for  V*  its  equivalent  2gh,  there  is  found 

V*  —  2V0U  +  2&2  -|-  2gkl 

~^h~ 

The  value  of  u  which  renders  e  a  maximum  is  by  the  usual 
method  found  to  be 

«  =  ibo  ; 

or  the  velocity  of  the  wheel  should  be  one-half  that  of  the  en-, 
tering  water.  Inserting  this  value,  the  efficiency  corresponding 
to  the  advantageous  velocity  is 


and  lastly,  replacing  v*  by  its  value  2gh^  ,  it  becomes 

i  /i.      h^ 

'=I-~;  ••••••  (9D 


340  HYDRAULIC  MOTORS.  [CHAP.  XI L 

which   is  the  theoretic   maximum  efficiency  of  the   overshot 
wheel. 

This  investigation  shows  that  one-half  of  the  entrance  fall 
h%  and  the  whole  of  the  exit  fall  /^  are  lost,  and  it  is  hence  plain 
that  in  order  to  make  e  as  large  as  possible  both  hQ  and  /^ 
should  be  as  small  as  possible.  The  fall  7/0  is  made  small  by 
making  the  radius  of  the  wheel  large  ;  but  it  cannot  be  zero,  for 
then  no  water  would  enter  the  wheel :  it  is  generally  taken  so 
as  to  make  the  angle  00  about  10  or  15  degrees.  The  fall  kl  is 
made  small  by  giving  to  the  buckets  a  form  which  will  retain 
the  water  as  long  as  possible.  As  the  water  really  leaves  the 
wheel  at  several  points  along  the  lower  circumference,  the  value 
of  hl  cannot  usually  be  determined  with  exactness. 

The  practical  advantageous  velocity  of  the  overshot  wheel, 
as  determined  by  the  method  of  Art.  124,  is  found  to  be  about 
o.4^0 ,  and  its  efficiency  is  found  to  be  high,  ranging  from  70  to 
90  per  cent.  In  times  of  drought,  when  the  water  supply  is 
low,  and  it  is  desirable  to  utilize  all  the  power  available,  its  effi- 
ciency is  the  highest,  since  then  the  buckets  are  but  partly 
filled  and  hl  becomes  small.  Herein  lies  the  great  advantage 
of  the  overshot  wheel ;  its  disadvantage  is  in  its  large  size  and 
the  expense  of  construction  and  maintenance. 

The  number  of  buckets  and  their  depth  are  governed  by  no 
laws  except  those  of  experience.  Usually  the  number  of  buck- 
ets is  about  ^r  or  6r,  if  r  is  the  radius  of  the  wheel  in  feet,  and 
their  radial  depth  is  from  10  to  15  inches.  The  breadth  of  the 
wheel  parallel  to  its  axis  depends  upon  the  quantity  of  water 
supplied,  and  should  be  so  great  that  the  buckets  are  not  fully 
filled  with  water,  in  order  that  they  may  retain  it  as  long  as 
possible  and  thus  make  /*,  small.  The  wheel  should  be  set  with 
its  outer  circumference  at  the  level  of  the  tail  water. 

Prob.  165.  Estimate  the  horse-power  of  an  overshot  wheel 
which  uses  1080  cubic  feet  of  water  per  minute  under  a  head 


ART.  140.] 


BREAST    WHEELS. 


341 


of  26  feet,  the  diameter  of  the  wheel  being  23  feet,  and  the 
water  entering  at  15°  from  the  top  and  leaving  at  12°  from  the 
bottom. 


ARTICLE  140.  BREAST  WHEELS. 

The  breast  wheel  is  applicable  to  small  falls,  and  the  action 
of  the  water  is  partly  by  impulse  and  partly  by  weight.  As 
represented  in  Fig.  98,  water  from  a  reservoir  is  admitted 
through  an  orifice  upon  the  wheel  under  the  head  h0  with  the 
velocity  v0 ;  the  water  being  then  confined  between  the  vanes 
and  the  curved  breast  acts  by  its  weight  through  a  distance  h^ , 


FIG.  98. 

which  is  approximately  equal  to  h  —  7z0 ,  until  finally  it  is  re- 
leased at  the  level  of  the  tail  race  and  departs  with  the  velocity 
?/,  which  is  the  same  as  that  of  the  circumference  of  the  wheel. 
The  total  energy  of  the  water  being  Wh,  the  work  of  the  wheel 
is  e  Wh,  if  e  be  its  efficiency. 

The  reasoning  of  the  last  article  may  be  applied  to  the 
breast  wheel,  h^  being  made  equal  to  zero,  and  the  expression 
there  deduced  for  e  may  be  regarded  as  an  approximate  value 
of  its  theoretic  efficiency.  It  appears,  then,  that  e  will  be  the 
greater  the  smaller  the  fall  /i0 ;  but  owing  to  leakage  between 


342  HYDRAULIC  MOTORS.  [CHAP.  XII. 

the  wheel  and  the  curved  breast,  which  cannot  be  theoretically 
estimated,  and  which  is  less  for  high  velocities  than  for  low 
ones,  it  is  not  desirable  to  make  VQ  and  /i0  small.  The  efficiency 
of  the  breast  wheel  is  hence  materially  less  than  that  of  the 
overshot,  and  usually  ranges  from  50  to  80  per  cent,  the  lower 
values  being  for  small  wheels. 

Another  method  of  determining  the  theoretic  efficiency  of 
the  breast  wheel  is  to  discuss  the  action  of  the  water  in  enter- 
ing and  leaving  the  vanes  as  a  case  of  impulse.  Let  at  the 
point  of  entrance  Av0  and  Au  be  drawn  parallel  and  equal  to 
the  velocities  v0  and  &,  the  former  being  that  of  the  entering 
water  and  the  latter  that  of  the  vanes.  Then  the  dynamic 
pressure  exerted  by  the  water  in  entering  upon  and  leaving 
the  vanes  is,  from  Art.  133, 

p  =  ^ (v  cos  a  _  u) 

-  g  n c 

and  the  work  performed  by  it  per  second  is 

W  t 

kQ=  —  (VQ  cos  a  —  mm . 
g  V 

This  is  a  maximum  when 

and  the  corresponding  work  of  the  impulse  is 

W 
kQ  =  —  v*  cos  ex . 

Adding  this  to  the  work  Wh^  done  by  the  weight  of  the  water, 
the  total  work  of  the  wheel  when  running  at  the  advantageous 
velocity  is 

k=  W 


ART.  141.]  UNDERSHOT    WHEELS.  343 

or  if  v*  be  replaced  by  its  value  c?  .  2ghQ  ,  where  cl  is  the  coeffi- 
cient of  velocity  as  determined  by  the  rules  of  Chapters  IV 
and  VI, 


whence  the  theoretic  efficiency  is 


If  in  this  expression  7*2  be  replaced  by  /i  —  /t0,  and  if  ^  =  I 
and  oc  =  o°,  this  reduces  to  the  same  value  as  found  for  the 
overshot  wheel.  The  angle  a,  however,  cannot  be  zero,  for 
then  the  direction  of  the  entering  water  would  be  tangential  to 
the  wheel,  and  it  could  not  impinge  upon  the  vanes;  its  value, 
however,  should  be  small,  say  from  10°  to  25°.  The  coefficient 
^  is  to  be  rendered  large  by  making  the  orifice  of  discharge 
with  well-rounded  inner  corners  so  as  to  avoid  contraction  and 
the  losses  incident  thereto.  The  above  formulas  cannot  be  re- 
lied upon  in  practice  to  give  close  values  of  k  and  e,  on  account 
of  losses  by  foam  and  leakage  along  the  curved  breast,  which 
of  course  cannot  be  algebraically  expressed. 

Prob.  166.  A  breast  wheel  is  10.5  feet  in  diameter,  and  has 
cv  =  0.93,  //„  =  4.2  feet,  and  a  =  12  degrees.  Compute  the 
most  advantageous  number  of  revolutions  per  minute. 

ARTICLE  141.  UNDERSHOT  WHEELS. 

The  common  undershot  wheel  has  plane  radial  vanes,  and 
the  water  passes  beneath  it  in  a  direction  nearly  horizontal. 
It  may  then  be  regarded  as  a  breast  wheel  where  the  action  is 
entirely  by  impulse,  so  that  in  the  preceding  equations  7/a  be- 
comes o,  h0  becomes  h,  and  a  will  be  o°.  The  theoretic  effi- 
ciency then  is 

<  =  *,' (92)' 


344  HYDRAULIC  MOTORS.  [CHAP.  XII. 

In  the  best  constructions  ct  is  nearly  unity,  so  that  it  may  be 
concluded  that  the  maximum  efficiency  of  the  undershot  wheel 
is  about  0.5.  Experiment  shows  that  its  actual  efficiency  varies 
from  0.20  to  0.40,  and  that  the  advantageous  velocity  is  about 
o.4^0  instead  of  0.5^.  The  lowest  efficiencies  are  obtained  from 
wheels  placed  in  an  unlimited  flowing  current,  as  upon  a  scow 
anchored  in  a  stream;  and  the  highest  from  those  where  the 
stream  beneath  the  wheel  is  confined  by  walls  so  as  to  prevent 
the  water  from  spreading  laterally. 

The  Poncelet  wheel,  so  called  from  its  distinguished  in- 
ventor, has  curved  vanes,  which  are  so  arranged  that  the  water 
leaves  them  tangentially,  with  its  absolute  velocity  less  than 
that  of  the  velocity  of  the  wheel.  If  in  Fig.  98  the  fall  h^  be 
very  small,  and  the  vanes  be  curved  more  than  represented,  it 
will  exhibit  the  main  features  of  the  Poncelet  wheel.  The 
water  entering  with  the  absolute  velocity  VQ  takes  the  velocity 
u  of  the  vane  and  the  velocity  V  relative  to  the  vane.  Passing 
then  under  the  wheel,  its  dynamic  pressure  performs  work;  and 
on  leaving  the  vane  its  relative  velocity  V  is  probably  nearly 
the  same  as  that  at  entrance.  Then  if  V  be  drawn  tangent  to 
the  vane  at  the  point  of  exit,  and  u  tangent  to  the  circumfer- 
ence, their  resultant  will  be  vl ,  the  absolute  velocity  of  exit, 
which  will  be  much  less  than  u.  Consequently  the  energy  car- 
ried away  by  the  departing  water  is  less  than  in  the  usual  forms 
of  breast  and  undershot  wheels,  and  it  is  found  by  experiment 
that  the  efficiency  may  be  as  high  as  60  per  cent. 

When  water  is  delivered  through  a  nozzle  against  the  vanes 
of  a  wheel  in  a  direction  nearly  tangent  to  the  lowest  point  of 
the  circumference,  the  action  is  entirely  similar  to  that  of  the 
undershot  wheel.  Such  machines  are  called  vertical  impulse 
wheels,  or  (improperly)  impact  wheels ;  they  are  extensively 
used  in  California  under  the  name  of  "  hurdy-gurdy  wheels," 
which  are  arranged  so  as  to  be  easily  transported  from  place 


ART.  142.]  HORIZONTAL  IMPULSE    WHEELS.  345 

to  place,  as  may  be  necessary  in  mining  operations,  the  nozzle 

being  attached  to  a  hose  or  pipe  which  brings  the  water  from 

a  canal.    Fig.  99  shows  an  outline  sketch  of  such  a  wheel.    The 

simplest  vanes  are  radial  planes  as  at  ^,but  these  are  found  to 

give  a   low  efficiency.     Curved  vanes, 

as  at  B,  are  also  used  with  much  better 

results,  as  they  cause  the  water  to  turn 

backward  opposite  to  the  direction  of 

the  motion,  and  thus  to  leave  with  a 

low  absolute  velocity  (Art.  1 34).     In  the 

plan  of  the  wheel  it  is  seen  that  the 

vanes  may  be  arranged  so  as  to  turn 

the    water    sidewise    while    deflecting 

it   backward.     The  experiments   made  FIG.  99. 

by  BROWNE*  show  that  with  plane  radial  vanes  the  highest 

efficiency  was  40.2  per  cent,  while  with  curved  vanes  or  cups 

82.5  per  cent  was  obtained.     The  velocity  of  the  vanes  which 

gave  the  highest  efficiency  was  in  each  case  found  to  be  a  little 

less  than  one-half  the  velocity  of  the  jet.     The  hurdy-gurdy  is 

a  fast-running  wheel,  as  the  stream  issuing  from  the  nozzle  has 

commonly  a  high  velocity. 

Prob.  167.  The  diameter  of  a  hurdy  gurdy  wheel  is  12.58 
feet  between  centres  of  vanes,  and  the  impinging  jet  has  a 
velocity  of  58.5  feet  per  second  and  a  diameter  of  0.182  feet. 
The  efficiency  of  the  wheel  is  44.5  per  cent  when  making  62 
revolutions  per  minute.  What  horse-power  does  it  furnish? 

Ans.  4.09  H.  P. 

ARTICLE  142.  HORIZONTAL  IMPULSE  WHEELS. 

Figure  100  represents  portions  of  the  circumference  of  two 
horizontal  wheels,  driven  by  the  dynamic  pressure  of  a  stream 
of  water  issuing  from  a  nozzle.  In  one  the  water  enters  the 

*  BOWIE'S  Practical  Treatise  on  Hydraulic  Mining,  p.  193. 


34^ 


HYDRAULIC  MOTORS. 


[CHAP.  XII. 


wheel  upon  the  inner  and  leaves  it  upon  the  outer  circumfer- 
ence, and  in  the  other  the  reverse  is  the  case.  The  first  form 
is  hence  called  an  outward-flow,  and  the  second  an  inward-flow, 
wheel.  The  water  issuing  from  the  nozzle  with  the  velocity  v  im- 
pinges upon  the  vanes,  and  in  passing  through  the  wheel  alters 
both  its  direction  and  its  absolute  velocity,  thus  transform- 
ing its  energy  into  useful  work.  The  energy  of  the  entering 

water  is  W — ,  and  that  of  the  departing  water  is  W— L,  if  vl  be 


FIG. 


its  absolute  velocity.     The  work  imparted  to  the  wheel  then  is 


2g          2g 

and  dividing  this  by  the  theoretic  energy,  the  efficiency  is 


e  —  I  —  -4-. 

This  is  the  same  as  the  general  formula  (90)  if  h'  =  o,  that  is, 
if  losses  in  foam  and  friction  are  disregarded,  and  if  the  wheel 
is  set  at  the  level  of  the  tail  race.  It  is  now  required  to  find  an 
expression,  for  i>,3,  whose  discussion  will  determine  the  condi- 
tions for  securing  the  greatest  efficiency.  The  reasoning  will 
be  general  and  applicable  to  both  outward-  and  inward-flow 
wheels. 

At  the  point  A  where  the  water  enters  the  wheel  let  the 


ART.  142.]  HORIZONTAL  IMPULSE    WHEELS.  347 

parallelogram  of  velocities  be  drawn,  the  absolute  velocity  of 
entrance  being  resolved  into  its  two  components,  the  velocity 
u  of  the  wheel  at  that  point,  and  the  velocity  F  relative  to  the 
vane  ;  let  ex.  be  the  angle  between  u  and  v,  and  0  be  the  angle 
between  u  and  V.  At  the  point  B  where  the  water  leaves  the 
wheel  let  F,  be  its  velocity  relative  to  the  vane,  and  ut  the 
velocity  of  the  wheel  at  that  point ;  then  their  resultant  is  vt , 
the  absolute  velocity  of  exit.  Let  ft  be  the  angle  between  V^ 
and  the  reverse  direction  of  ul ,  and  6  the  angle  between  u^  and 
TV  The  directions  of  the  velocities  u  and  #,  are  of  course  tan- 
gential to  the  circumferences  at  the  points  A  and  B.  Let  r 
and  rl  be  the  radii  of  these  circumferences ;  then  the  velocities 
of  revolution  are  directly  as  the  radii,  or  url  =  up. 

Now  to  determine  the  value  of  v*  the  triangle  at  It  between 
ul  and  z/j  gives 

v*  =  u?  +  V?  -  2ulVl  cos  ft. 

The  value  of  Vl  is  found  by  the  formula  (88)  of  Art.  136, 
namely, 

V?  =  V*  -  u*  +  u,\ 

The  value  of  F2  from  the  triangle  at  A  between  u  and  v  is 

Fa  =  //  +  v1  —  2uv  cos  a. 
Hence  the  first  equation  becomes 


v*=  v*  —  2uv  cos  a  -f-  2u*—  2ul  cos  ft  Vv*  —  2uv  cos  a  -(-  u*. 

Substituting  for  2il  its  value  in  terms  of  u,  and  placing  the 
resultant  value  of  v?  in  the  expression  for  the  efficiency,  there 
is  found 

u  if      r* 

e  =  2  —  cos  ex  —  2-r  .  — „ 

v  v        r 


~2^cosa+--  •  (93) 

This  is  the  general  formula  for  the  efficiency  of  a  horizontal- 


34$  HYDRAULIC  MOTORS.  [CHAP.  XII. 

impulse  wheel,  and  it  will  now  be  discussed  in  order  to  deter- 
mine what  values  of  or,  ft,  u>  and  — '  are  most  advantageous. 


The  value  of  u  which  renders  e  a  maximum  cannot  be  de- 
termined in  a  form  which  is  available  for  use,  since  the  deriva- 
tive equated  to  zero  leads  to  a  biquadratic  equation.  But 
when  cos  ft  =  o,  or  ft  =  90°,  the  advantageous  velocity  is 

i    r* 
u  = 2 v  cos  a, 

as  also  shown  in  Art.  136,  and  for  an  inward-flow  wheel  ^can  be 
made  nearly  unity.  It  is  also  seen  from  Fig.  100  that  the  only 
way  in  which  vt  can  be  made  o  is  when  u^  =  V^  and  ft  =  o°.  It 
is  hence  a  good  rule  that  ft  should  be  a  small  angle,  but  this  is 
more  important  in  an  outward-  than  in  an  inward-flow  wheeL 
The  formula  shows  also  that  e  increases  when  cos  ft  and  cos  a 
increase,  and  hence  in  general  both  ft  and  a  ought  to  be  small 
angles. 

If  the  angle  ft  be  not  large,  the  condition  #,  =  V^  will  give 
a  near  approach  to  the  conditions  of  best  efficiency.  When 
this  occurs,  u  =  V,  and  the  advantageous  velocity  is 


u  = 


2  cos  a 

This  reduces  the  formula  for  the  efficiency  to 

i    r*  i  -  cos  ft 


It  is  clearly  shown  by  this  formula  that  in  an  inward-flow 
wheel,  where  rl  is  small  compared  with  r,  e  may  have  a  high 
value  even  if  a  and  ft  are  not  very  small  angles.  For  instance, 
let  r  =  $rlta  =  30°,  and  ft  =  45°;  then 

e  =  i  —  0.5  X  0.04  X  f  (i  —  0.707)  =  0.992. 


ART.  143.]  REACTION    WHEELS.  349 

For  an  outward-flow  wheel,  however,  where  ^  is  larger  than  rt 
it  is  absolutely  necessary  that  ft  should  be  small. 

The  actual  efficiency  of  horizontal-impulse  wheels  is  rarely 
greater  than  75  per  cent.  There  is,  of  course,  a  material  loss 
of  energy  in  foam  when  the  water  enters  the  wheel,  and  in 
friction  as  it  passes  through  it.  To  reduce  the  foam  as  much 
as  possible,  the  direction  of  the  vanes  at  the  entrance  circum- 
ference should  correspond  with  the  relative  velocity  V.  If  the 
advantageous  velocity  u  is  known,  the  angle  0  which  deter- 
mines this  direction  is  given  by  the  rule  established  in  Art.  133, 
namely, 

u 

cot  0  =  cot  a : . 

v  sin  a 

"V 

If  the  wheel  be  run  at  the  velocity  u  =  ,  this  becomes 

2  cos  a 

cot  0  =  cot  2a, 

and  hence  0  is  double  the  angle  a.  It  is  usual  to  make  0 
somewhat  greater  than  2a,  however,  and  it  is  often  made  90° 
when  a  is  less  than  30°.  For  this  practice  there  seems  to  be 
no  good  reason. 

Prob.  168.  If  the  wheel  be  run  at  such  a  velocity  that  vl  is 
radial,  or  6  =  90°,  deduce  expressions  for  u  and  e,  and  show 
that  the  efficiency  is  less  than  in  the  above  case  where  «t  =  Vl . 


ARTICLE  143.  REACTION  WHEELS. 

The  reaction  wheel,  sometimes  called  Barker's  mill,  con- 
sists of  a  number  of  hollow  arms  connected  with  a  hollow  ver- 
tical shaft,  as  shown  in  Fig.  101.  The  water  issues  from  the 
ends  of  the  arms  in  a  direction  opposite  to  that  of  their  motion, 
and  by  the  dynamic  pressure  due  to  its  reaction  the  energy  of 
the  water  is  transformed  into  useful  work.  Let  the  head  of 


350 


HYDRAULIC  MOTORS. 


[CHAP.  XIL 


water  CC  in  the  shaft  be  h ;  then  the  pressure-head  BB  which 

causes  the  flow  from  the  arms  is  greater 

>>  than  ^,  on  account  of  centrifugal  force. 
Let  Fj  be  the  velocity  of  discharge 
relative  to  the  wheel ;  then,  as  shown  in 

Art.  29,  

^=1 


i    i 


The  absolute  velocity  v^  of  the  issuing 
water  now  is 


It  is  seen  at  once  that  the  efficiency 
can  never  reach  unity  unless  7^  —  o, , 
which  requires  that  F,  =  #, .  This, 
however,  can  only  occur  when  u^  =  oo , 
since  the  above  formula  shows  that  Ft 
must  be  greater  than  ul  for  any  finite 
values  of  h  and  ul .  To  deduce  an  ex- 
pression for  the  efficiency  the  general  formula  (90)  may  be 
used,  placing  for  v?  its  value  (F,  —  uy,  and  for  v*  its  value  2gk 
or  V?  —  u?  ;  then 

e  —  l          fTi        771  =  77~i~rr (94) 


Fie.  101. 


This  shows,  as  before,  that  e  equals  unity  when  Fz  =  uI  =  oo. 
If  Fj  =  2«j  the  value  of  ^  is  0.667;  if  Fx  =  3«x,  the  value  of  e  is 
0.50. 

The  above  formula  may  be  deduced  in  another  way,  as 
follows  :  The  absolute  velocity  of  the  issuing  water  being 
F!  —  ul  ,  the  dynamic  pressure  of  its  reaction  is 

W  W 


and  the  work  done  by  this  pressure  is 

W 

^=Pu1  =  —(Fl 

o 


ART.  143.]  REACTION    WHEELS.  351 

The  theoretic  energy  of  the  water  is 


and  accordingly  the  efficiency  is 

k  2u, 

'^x^r^  ........  (94) 

This  investigation  apparently  shows,  although  e  approaches 
unity  as  ul  approaches  V^  ,  that  the  effective  work  k  decreases. 
This  is  due  to  the  fact  that  W,  as  above  written,  is  regarded  as 
constant,  which  is  not  the  case,  as  its  value  is  wa1  V^  .  Hence, 
really,  as  z/x  increases  so  does  W,  and  when  u1  =  V^  =  oo,  the 
value  of  k  is  Wh  ;  but  as  W  is  then  also  oo  ,  the  work  would  be 
unlimited.  Practically,  of  course,  none  of  these  conditions  can 
be  approached,  so  that  the  full  theoretic  efficiency  of  the  reac- 
tion wheel  can  never  be  realized. 

To  consider  the  effect  of  friction  in  the  arms,  let  cl  be  the 
coefficient  of  velocity  (Chapter  VI),  so  that 


V,   =  C,  V2gk  +  U?. 

Then  the  effective  work  of  the  wheel  is 

W  

k  =  —(w  V2gh+u?  -  «/), 

and  the  efficiency  is 


The  value  of  ul  which  renders  this  a  maximum  is 

g* 


tL     = 


and  this  reduces  the  value  of  the  efficiency  to 

e  =  i  ._  4/1  -  c? (94)' 

If  cl  —  I,  there  is  no  loss  in  friction,  and  «t  =  oo  and  e  =  I, 
as  before  deduced.  If  cl  =  0.94,  the  advantageous  velocity  ul 
is  very  nearly  V2gh,  and  e  is  0.66 ;  hence  the  influence  of  fric- 


352  HYDRAULIC  MOTORS.  [CHAP.  XII. 

tion  in  diminishing  the  efficiency  is  very  great.  In  order  to 
make  cl  large,  the  end  of  the  arm  where  the  water  enters  must 
be  well  rounded  to  prevent  contraction,  and  the  interior  surface 
must  be  smooth.  If  the  inner  end  has  sharp  square  edges,  as 
in  a  standard  tube  (Art.  61),  cl  is  0.82,  and  e  becomes  about 
0-43- 

The  reaction  wheel  is  not  now  used  as  a  hydraulic  motor 
on  account  of  its  low  efficiency.  Even  when  run  at  high  speeds 
the  efficiency  is  low  on  account  of  the  greater  friction  and 
resistance  of  the  air.  By  experiments  on  a  wheel  one  meter  in 
diameter  under  a  head  of  1.3  feet  WEISBACH*  found  a  maxi- 
mum efficiency  of  67  per  cent  when  the  velocity  of  revolution 
u^  was  V2gh.  When  &x  was  2  Vzgh  the  efficiency  was  nothing, 
or  all  the  energy  was  consumed  in  frictional  resistances.  By 
reasoning  like  that  of  Art.  137  it  may  be  shown  when  the  issu- 
ing water  makes  an  angle  /?  with  the  line  of  motion,  as  in  Fig. 

96,  that  

e  =  j  _  .  Vi  -  c?  cos2  A (94)" 

and  that  the  corresponding  advantageous  speed  is  given  by  the 
formula  above  if  cl  be  replaced  by  c1  cos  ft. 

If  the  water  issuing  from  the  arms  impinge  upon  planes 
firmly  attached  to  the  arms,  as  at  M  in  Fig. 
102,  no  motion  occurs,  for  the  water  is  then 
deflected  in  a  radial  direction,  and  it  exerts 
no  dynamic  pressure  which  can  cause  revolt 
tion.  If,  however,  it  impinge  in  hemispherical 
cups  attached  to  the  arms  which  deflect  the 
water  backward,  the  motion  occurs  in  a  con- 
trary direction  to  that  of  the  common  reac- 
FIG.  102.  tion  wheel,  for  the  velocity  ul  must  be  oppo- 

site to  the  absolute  velocity  Vl  —  z/,  .     Lastly,  it  may  be  men- 

*  Hydraulics  and  Hydraulic  Motors,  DuBois's  translation,  p.  385. 


ART.  144.]         FLO IV   THROUGH   TURBINE    WHEELS.  353 

tioned  that  the  common  reaction  wheel  when  placed  under 
water  sometimes  runs  in  a  direction  opposite  to  that  of  its 
usual  motion.  This  is  probably  due  to  the  circumstance  that 
there  is  no  static  pressure  in  front  of  the  orifice  where  the 
stream  issues;  so  that,  if  the  depth  of  immersion  be  sufficient, 
the  static  pressure  behind  it  may  be  greater  than  the  dynamic 
pressure  of  the  reaction,  and  accordingly  the  resultant  pressure 
is  in  the  direction  of  the  flow.  There  are,  in  fact,  few  machines 
which  illustrate  so  many  of  the  laws  of  hydraulics  as  this  mar- 
vellous reaction  wheel. 

Prob.  169.  Compute  the  effective  power  of  a  reaction  wheel 
when  ft  =  o°,  h  =  16  feet,  ul  —  V2ghy  c1  =  0.95,  rt  =  1.75  feet, 
and  al  =  4.25  square  inches,  the  latter  being  the  sum  of  the 
areas  of  the  exit  orifices.  Ans.  1.59  H.P.  • 


ARTICLE  144.  FLOW  THROUGH  TURBINE  WHEELS. 

A  turbine  wheel  acts  under  the  dynamic  pressure  of  flow- 
ing water  which  at  the  same  time  may  be  under  a  certain  de- 
gree of  static  pressure.  If  in  the  reaction  wheel  of  Fig.  96 
the  arms  be  separated  from  the  penstock  at  A,  and  be  so  ar- 
ranged that  BA  revolves  around  the  axis  while  AC  is  station- 
ary, the  resulting  apparatus  may  be  called  a  turbine.  The 
static  pressure  of  the  head  CC  can  still  be  transmitted  through 
the  arms,  so  that,  as  in  the  reaction  wheel,  the  discharge  will  be 
influenced  by  the  speed  of  rotation.  The  general  arrangement 
of  entrance  and  exit  angles,  however,  is  like  the  impulse  wheel, 
the  dynamic  pressure  being  due  to  reaction  only  in  a  slight 
degree.  Turbine  wheels  are  now  used  more  extensively  than 
other  hydraulic  motors,  on  account  of  their  cheapness,  com- 
pactness, and  high  efficiency.  The  different  kinds  may  be 
classified  as  outward  flow,  inward  flow,  downward  flow,  and 
those  in  which  the  flow  is  partly  inward  and  partly  downward. 


354 


HYDRAULIC  MOTORS. 


[CHAP.  XII. 


In  Fig.  103  are  shown  horizontal  and  vertical  sections  of  the 
outward-  and  inward-flow  types,  without  the  inclosing  case. 
The  moving  wheel  marked  W  consists  of  a  series  of  vanes  set 
in  a  frame  which  is  attached  by  arms  to  the  central  axis  ;  the 
spaces  between  these  vanes  will  be  called  buckets.  The  water 
is  brought  to  the  buckets  by  a  series  of  guides  set  in  a  fixed 
frame  G,  and  it  is  seen  that  the  water  is  introduced  around  the 
entire  circumference  of  the  wheel.  Between  the  guides  and 
the  wheel  is  an  annular  space  in  which  slides  an  annular  verti- 


FIG.  103. 

cal  gate ;  this  stops  the  admission  of  water  when  entirely  de- 
pressed, and  serves  to  regulate  the  quantity  furnished.  The 
spaces  between  the  guides,  as  also  the  buckets,  are  usually  en- 
tirely filled  with  water  when  the  wheel  is  in  motion,  and  such 
will  be  taken  to  be  the  case  in  the  following  discussions. 

A  formula  for  the  discharge  q  through  a  turbine  wheel 
when  the  gate  is  fully  raised  is  now  to  be  established.  Let  h 
be  the  head  between  the  water  levels  in  the  penstock  and  tail 
race,  Hl  the  pressure-head  on  the  exit  orifices  or  the  depth  of 
the  latter  below  the  tail  water  level,  and  H  the  pressure-head 


ART.  I44-]         FLOW   THROUGH   TURBINE    WHEELS. 


355 


at  the  gate  opening  as  indicated  by  a  piezometer  supposed  to 
be  there  inserted  as  seen  in  Fig.  104. 
Let  ul  and  u  be  the  velocities  of  the 
wheel  at  the  exit  and  entrance  circum- 
ference, whose  radii  are  rl  and  r  (Fig. 
103).  Let  Vl  and  V  be  the  relative 
velocities  of  exit  and  entrance,  and  v9 
be  the  absolute  velocity  of  the  water  — 
as  it  leaves  the  guides  and  enters  the 
wheel  ;  v0  may  be  less  or  greater  than 
,  depending  upon  the  value  of  the 


FIG.  io4. 

pressure-head  H.  Let  a^  ,  a,  and  a0  be  the  areas  of  the  orifices 
normal  to  the  directions  of  Vl  ,  V,  and  v0  .  Now,  neglecting  all 
losses  of  friction  between  the  guides,  the  theorem  of  Art.  27, 
that  pressure-head  plus  velocity-head  equals  the  total  head, 
gives 


Also,  neglecting  the  friction  and  foam  in  the  buckets,  the  the- 
orem of  Art.  137  gives 

H  +^--^--H+  —  -- 

° 


Adding  these  equations,  the  pressure-heads  //,  and  H  disap- 
pear  and  there  results  the  formula 


V   - 


.'  =  2gh  +  u:  -  u\ 


Now,  since  the  buckets  are  fully  filled,  the  same  quantity  of 
water,  q,  passes  in  each  second  through  each  of  the  areas  a^  ,  a, 

and  a0  ,  and 


Introducing  these  values  of  the  velocity,  solving  for  ^,  and 


HYDRAULIC  MOTORS.  [€HAP.  XIL 

multiplying  by  a  coefficient  c  to  account  for  losses  in  leakage 
and  friction,  the  discharge  per  second  is 


This  is  the  formula  for  the  flow  through  a  turbine  in  motion, 
when  the  gate  is  fully  raised.  In  an  outward-flow  turbine  ul 
is  greater  than  u,  and  consequently  the  discharge  increases 
with  the  speed  ;  in  an  inward-flow  turbine  u^  is  less  than  u,  and 
consequently  the  discharge  decreases  as  the  speed  increases. 

The  value  of  the  coefficient  c  will  probably  vary  with  the 
head,  and  also  with  the  size  of  the  areas  al ,  a,  and  a0 .  For  the 
outward-flow  Boyden  turbine,  the  tests  of  which  are  given 
in  Art.  126,  it  lies  between  0.94  and  0.95,  as  the  following 
results  show,  where  the  first  four  columns  contain  the  number 
of  the  experiment,  the  observed  head,  number  of  revolutions 
per  minute,  and  discharge  in  cubic  feet  per  second.  The  fifth 
column  gives  the  theoretic  discharge  computed  from  the  above 
formula,  taking  the  coefficient  as  unity,  and  the  last  column  is 


No. 

*. 

N. 

fr 

Q. 

c. 

21 

17.16 

63.5 

117.01 

123.1 

0.950 

20 

17.27 

70.0 

118.37 

125.2 

0.945 

19 

17.33 

75.0 

119-53 

126.8 

0-943 

18 

17.34 

8o.b 

121.15 

128.4 

0.944 

17 

17.21 

86.0 

122.41 

130.0 

0.942 

16 

17.21 

93-2 

124.74 

132.5 

0.941 

15 

17.19 

IOO.O 

127.73 

134-9 

0.947 

derived  by  dividing  the  observed  discharge  q  by  the  theoretic 
discharge  Q.  The  discrepancy  of  5  or  6  per  cent  is  smaller 
than  might  be  expected,  since  the  formula  does  not  consider 
frictional  resistances. 

A  satisfactory  formula  for  the  discharge  through  a  turbine 


ART.  144.]         FLOW   THROUGH   TURBINE    WHEELS.  357 

when  the  gate  is  partly  depressed  is  difficult  to  deduce,  because 
the  loss  of  head  which  then  results  can  only  be  expressed  by 
the  help  of  experimental  coefficients  similar  to  those  given  in 
Art.  75  for  the  sliding  gate  in  a  water  pipe,  and  the  values  of 
these  for  turbines  are  not  known.  It  is,  however,  certain  that 
for  each  particular  gate  opening  the  discharge  is  given  by 


q  =  m  V2gk  +  u?  -  if ; (95)' 

in  which  m  depends  upon  the  areas  of  the  orifices  and  the  height 
to  which  the  gate  is  raised.  For  instance,  in  the  tests  of  the 
Boyden  turbine  of  Art.  126,  the  value  of  ;;/  is  2.815  when  the 
proportional  gate  opening  is  0.609,  and  the  computed  discharges 
will  differ  in  no  case  more  than  one  per  cent  from  those 
observed ;  when  the  proportional  gate  opening  is  0.200,  the 
value  of  m  is  1.357.  And  each  turbine  will  have  its  own  values 
of  m,  depending  upon  the  area  of  its  orifices. 

A  downward-flow  turbine  is  one  in  which  each  particle  of 
water  remains  at  the  same  distance  from  the  axis  in  its  path 
through  the  guides  and  buckets.  In  Fig.  105  is  seen  a  semi- 
vertical  section  of  the  wheel,  and  also  a  development  of  a  por- 


FIG.  105. 

tion  of  a  cylindrical  section  showing  the  inner  arrangement. 
The  formula  for  the  discharge  can  be  adapted  to  this  by  mak- 
#!  =  u.  In  this  turbine  there  is  no  action  of  centrifugal  force, 
so  that  the  relative  exit  velocity  Vl  is  equal  to  V,  or  at  the 
most  equal  to  VV  -f-  2ghQ ,  where  //„  is  the  vertical  depth  of  the 
wheel. 


HYDRAULIC  MOTORS.  [CHAP.  XII. 

The  three  typical  classes  of  turbines  above  described  are 
•often  called  by  the  names  of  those  who  first  invented  or  per- 
fected them  ;  thus  the  outward-flow  is  called  the  Fourneyron, 
the  inward-flow  the  Francis,  and  the  downward-flow  the  Jonval, 
turbine.  There  are  also  many  turbines  in  the  market  in  which 
the  flow  is  a  combination  of  inward  and  downward  motion,  the 
water  entering  horizontally  and  inward,  and  leaving  vertically ; 
the  bucket  partitions  in  these  are  warped  surfaces,  and  the 
angle  ft  is  not  the  same  at  all  points  of  exit.  These  wheels  of 
combined  direction  seem  on  the  whole  to  be  those  which  give 
the  promise  of  ultimately  attaining  the  highest  efficiency.  The 
usual  efficiency  of  turbines  at  full  gate  is  from  70  to  85  per 
cent,  although  90  per  cent  has  in  some  cases  been  derived. 
When  the  gate  is  partly  closed  the  efficiency  in  general  de- 
creases, and  when  the  gate  opening  is  small  it  becomes  very 
low,  as  the  example  in  Art.  126  shows.  This  is  due  to  the  loss 
of  head  consequent  upon  the  sudden  change  of  cross-section ; 
and  therein  lies  the  disadvantage  of  the  turbine,  for  when  the 
water  supply  is  low,  it  is  important  that  the  wheel  should 
utilize  all  the  power  available.  That  this  difficulty  will  ulti- 
mately be  overcome  there  can  be  little  doubt.* 

Prob.  170.  Deduce  the  value  of  the  coefficient  m  for  the 
experiments  Nos.  1-7  in  Art.  126,  and  then  compare  the  com- 
puted with  the  observed  discharges. 

ARTICLE  145.  THEORY  OF  TURBINES. 
A  volume  could  easily  be  written  upon  the  theory  of  tur- 
bines alone,  and  hence  the  brief  space  here  available  must  be 
limited  to  the  most  important  topic  among  the  many  which 
might  be  discussed,  namely,  the  conditions  of  maximum  effi- 
ciency. This  may  be  divided  into  two  problems:  first,  to 

*  Concerning  a  wheel  whose  efficiency  is  highest  at  about  three-fourths  gate, 
see  Transactions  American  Society  Mechanical  Engineers,  vol.  viii.  p.  359. 


ART.  I45-]  THEORY  OF   TURBINES.  359 

determine  the  advantageous  velocity,  and  the  corresponding 
efficiency,  for  a  turbine  whose  dimensions  are  known;  and 
second,  to  ascertain  how  a  turbine  should  be  built  so  that  the 
greatest  efficiency  can  be  obtained.  The  investigations  will  be 
limited  to  the  case  where  the  gate  is  fully  open.  In  both 
problems  there  are  four  sources  of  loss  of  energy  which  should 
be  carefully  kept  in  mind  :  first,  loss  in  the  absolute  velocity  of 
the  departing  water ;  second,  loss  in  friction  ;  third,  loss  in 
foam  ;  fourth,  loss  in  leakage  between  the  guides  and  the  wheel. 

The  first  problem  may  be  discussed  by  imposing  the  condi- 
tion that  the  absolute  velocity  vl  of  the  departing  water  should 
be  as  small  as  possible.  This  will  occur,  if  f3  be  not  a  large 
angle,  when  ul  =  Fa ,  as  then  their  resultant  v,  becomes  very 
small.  Now  V^  may  be  expressed  in  terms  of  the  discharge  q 
and  the  area  a, ;  thus  the  condition  for  the  minimum  is 


Substituting  the  value  of  q  from  (95),  and  solving  for  «,,  there 
is  found 

u*  =  —     — i '—} -^r—  - ,      ....     (96) 

°  ^       ^       <? 

which  gives  the  advantageous  velocity  of  the  circumference 
where  the  water  leaves  the  wheel.  When  u1  =  Vlt  the  velocity- 
square  of  the  departing  water  is 

v*  =  2u*  (i  —  cos  ft)  •=  4u^  sin2  %ft ; 
and  accordingly  the  efficiency  of  the  turbine  is 


in  which  the  last  term  represents  the  losses  due  to  friction,  foam, 


36O  HYDRAULIC  MOTORS.  [CHAP.  XII. 

and  leakage.  These  losses  can  scarcely  be  formulated,  and  as 
their  sum  is  often  greater  than  the  loss  due  to  the  energy  of  the 
departing  water,  which  is  represented  by  the  second  term,  it  is 
plain  that  the  determination  of  the  theoretic  efficiency  in  any 
particular  case  cannot  often  be  successfully  made. 

The  second  problem,  to  design  a  turbine  so  that  the  effi- 
ciency may  be  a  maximum  at  full  gate,  can  be  also  approxi- 
mately solved  by  making  ul  =  Vl,  and  by  adding  the  condition 
that  there  should  be  no  leakage.  Resuming  the  first  and  second 
equations  of  the  last  article,  it  is  seen  from  Fig.  104  that  there 
will  be  no  leakage  if  H  =  H^ .  The  two  conditions  hence  reduce 
the  second  equation  to  u  =  V,  and  the  first  to  v*  =  2gh.  But 
when  u  —  V,  the  angle  0  should  be  2«,  in  order  that  the  water 
may  enter  the  buckets  tangentially.  Also  q  =  al  Vl  =  a1u1 , 
and  q  =  aV  =  au\  hence  alul  equals  au,  or  alrl  equals  ar. 
Therefore,  to  prevent  leakage  and  foam,  the  turbine  should  be 
so  built  that 

0  =  2«,       and       a^  —  ar (97) 

These,  however,  give  maximum  efficiency  only  when  the  speed 
of  the  wheel  corresponds  to  the  condition  u  =  V\  inserting  for 
Fits  value  in  terms  of  z;0  and  or,  this  furnishes 

u  —  — ^ —  =  I  sec  a  ^2gJi ,  .     .     .     .     (97)' 

2  COS  a        2 

which  is  the  advantageous  velocity  of  the  circumference  where 
the  water  enters.  For  the  other  circumference 

¥  I    ¥ 

ul  =  u  —  —  -  —  sec  a  \/2gh , 
and  then  the  maximum  efficiency  is 

<?=  i-^sec'tfsin2^/?--,      .     .     .     (97)" 


ART.  145-1  THEORY  OF   TURBINES.  361 

in  which  the  last  term  refers  to  frictional  losses  only,  since 
leakage  and  foam  are  avoided  by  the  construction.  The  fric- 
tion of  the  water  on  the  guides  and  vanes  may  be  estimated 
from  general  knowledge  regarding  flow  in  tubes  to  consume 
about  5  per  cent,  and  for  the  axle  friction  there  may  be  put  2 
per  cent ;  thus  the  least  value  of  ti  is  probably  about  0.07/1. 

It  is  seen  from  the  preceding  discussions  that  the  efficiency 
increases  as  a  and  ft  decrease,  but  that  ft  is  more  important 
than  a ;  for  if  ft  in  (97}"  be  o,  the  angle  a  may  have  any  value 
less  than  a  right  angle,  and  the  term  containing  it  will  vanish. 
It  is  likewise  seen  that  e  increases  as  the  ratio  r,  -^  r  decreases, 
that  is,  an  inward-flow  turbine  is  preferable  to  one  of  outward 
flow ;  and  if  this  ratio  can  be  very  small,  the  angle  ft  need  not 
necessarily  be  small.  The  areas  a^ ,  a,  and  a0 ,  above  used,  are 
those  normal  to  the  directions  of  the  velocities  V^ ,  V,  and  ^ ; 
they  can  be  expressed  in  terms  of  the  radii  r^  and  r,  and  the 
vertical  depths  d^  and  d,  as  follows : 

al  =  2nrldl  sin  ft,     «  —  2nrd  sin  0,     aQ  =  2nrd  sin  a. 

Using  these  values,  the  second  condition  in  (97)  may  be  other- 
wise expressed  by 

r*dl  sin  ft  =  r*d  sin  2 a. 

From  this  equation,  after  assuming  the  angles  and  radii,  the 
depths  </,  and  d  can  be  arranged.  The  above  values  of  the 
areas  can  also,  if  desired,  be  inserted  in  the  formulas  (96)  and 
(97).  It  is  customary  in  testing  a  turbine  to  measure  these 
areas  directly,  but  in  making  a  design  the  above  expressions 
will  be  found  useful. 

Referring  again  to  the  outward-flow  turbine  of  Art.  126,  it 
is  seen  that  neither  of  the  conditions  of  (97)  are  fulfilled ;  for  0 
is  90°  and  a  is  24°,  while  a^r^  =  15.3  and  ar  =  32,3.  There  ex- 
isted, therefore,  a  loss  by  leakage  under  the  gate,  and  a  loss  due 


362  HYDRAULIC  MOTORS.  [CHAP.  XII. 

to  foam  and  enlargement  of  section  at  entrance  into  the  wheel. 
By  formula  (96)'  it  is  found  that  the  loss  due  to  the  energy 
of  the  departing  water  was  about  II  per  cent.  If  the  frictional 
losses  were  about  7  per  cent,  the  losses  due  to  leakage  and 
foam  amounted  to  about  5  per  cent,  since  the  observed  effi- 
ciency was  77  per  cent.  The  value  of  ul  in  experiment  20  was 
24.3  feet  per  second,  and  that  of  Vl  was  25.7,  so  that  the  con- 
dition «j  =  Vl  was  approximately  fulfilled.  For  experiment 
19,  the  value  of  ul  is  26.0  and  that  of  Vl  is  25.9,  so  that  for 
some  speed  intermediate  between  those  of  the  two  experiments 
the  condition  would  have  been  exactly  satisfied.  It  is  hence 
indicated  that  the  assumption  of  deducing  the  advantageous 
velocity  from  the  minimum  of  v*  is  entirely  correct. 

Prob.  171.  Show  that  in  experiment  20,  above  alluded  to, 
the  velocity  v0  was  24.9  feet  per  second ;  that  the  pressure-head 
H  —  Hl ,  which  caused  leakage,  was  7.64  feet. 


ARTICLE  146.  OTHER  KINDS  OF  MOTORS. 

Machines  in  which  the  static  pressure  of  water  acts  upon 
pistons  have  been  used  to  some  extent,  particularly  for  small 
motors,  although  in  Europe  under  the  name  of  water-pressure 
engines  many  large  ones  have  been  built.  Fig.  106  shows  in  a 
diagrammatic  way  the  method  of  their  action,  in  which  are  seen 
two  reservoirs  A  and  B,  the  head  between  them  being  //. 
When  the  valves  marked  M  are  open  and  those  marked  N  are 
closed  water  passes  from  the  upper  to  the  lower  reservoir,  and 
the  piston  moves  in  the  direction  of  the  arrow.  As  the  piston 
reaches  the  end  of  its  stroke,  the  valves  M  are  closed  while  N 
are  opened  ;  the  piston  then  reverses  its  motion,  and  water 
passes  from  A  to  B  through  the  other  set  of  pipes.  In  the 
practical  construction  of  the  apparatus  it  is  necessary  that  the 


ART.  146.] 


OTHER  KINDS   OF  MOTORS. 


3^3 


pipes  should  be  large,  so  that  the  velocity  of  flow  may  be  small 
in  order  to  render  the  fric- 

-^  A  • 

tional  resistances  as  slight  as 
possible,  and  to  avoid  the 
shocks  wh'ich  would  result 
when  the  valves  close.  The 
reservoir  B  may  be  omitted, 
so  that  the  water  from  the  cyl- 
inder discharges  through  the 
lower  valves  directly  into  the 
air,  and  arrangements  may  be  FIG.  106. 

devised  so  that  but  one  pipe  from  the  upper  reservoir  is 
needed.  In  any  event  the  principle  of  action  is  like  that  of 
the  steam-engine,  except  that  there  can  be  no  cut-off  until  the 
piston  reaches  the  end  of  its  stroke.  If  frictional  resistances 
could  be  entirely  avoided,  the  theoretic  efficiency  of  this  motor 
might  be  made  very  high,  for  the  work  done  in  one  stroke  is 
wA/i  if  A  be  the  area  of  the  piston,  and  in  one  second  is  nwAh 
if  n  be  the  number  of  strokes  per  second.  But  nwA  is  the 
weight  of  water  delivered  per  second,  and  thus  the  work  done 
is  W/i,  which  is  the  theoretic  energy  of  the  fall. 

The  screw  wheel  consists  of  one  or  two  turns  of  a  helicoidal 
surface  around  a  vertical  shaft,  the  screw  being  inclosed  in  a 
cylindrical  case  which  prevents  the  water  from  escaping.  The 
downward  pressure  of  the  water  can  then  be  resolved  into  two 
components :  one  parallel  to  the  surface,  which  causes  a  relative 
velocity  V\  and  one  horizontal,  which  corresponds  to  the 
velocity  of  the  wheel.  This  apparatus  has  never  come  into 
practical  use  as  a  motor,  and  probably  for  the  reason  that,  like 
the  reaction  wheel,  an  infinite  velocity  of  revolution  is  theo- 
retically necessary  in  order  to  secure  maximum  efficiency. 

The  old-style  tub  wheel  is  a  horizontal  impulse  wheel  which 
consists  of  a  series  of  buckets  into  which  a  stream  from  a  spout 


364  HYDRAULIC  MOTORS.  [CHAP.  XI L 

impinges  in  an  inclined  direction.  They  are  not  efficient,  and 
hence  are  now  little  used.  The  same  remark  may  be  made 
concerning  many  other  devices  which  have  from  time  to  time 
been  used  as  hydraulic  motors.  The  wheels  described  in  the 
preceding  pages  are  almost  the  only  ones  now  in  use,  and  cer- 
tainly the  only  ones  whose  use  is  extensive.  The  turbine  now 
leads  all  the  others  on  account  of  its  small  size,  cheapness,  ap- 
plicability to  both  large  and  small  falls,  and  high  efficiency.  In 
this  improvements  are  constantly  being  made,  and  undoubtedly 
it  is  the  wheel  of  the  future. 

Prob.  172.  How  can  a  turbine  be  set  30  feet  above  the  level 
of  the  tail  race,  and  still  secure  the  power  due  to  the  total 
head? 


ART.  147.]  GENERAL  PRINCIPLES.  365 


CHAPTER  XIII. 
NAVAL   HYDROMECHANICS. 

ARTICLE  147.  GENERAL  PRINCIPLES. 

In  this  chapter  is  to  be  discussed  in  a  brief  and  elementary 
manner  the  subject  of  the  resistance  of  water  to  the  motion  of 
vessels,  and  the  general  hydrodynamic  principles  relating  to 
their  propulsion.  The  water  may  be  at  rest  and  the  vessel  in 
motion — or  both  may  be  in  motion,  as  in  the  case  of  a  boat  go- 
ing up  or  down  a  river.  In  either  event  the  velocity  of  the 
vessel  relative  to  the  water  need  only  be  considered,  and  this 
will  be  called  v.  The  simplest  method  of  propulsion  is  by  the 
oar  or  paddle;  then  come  the  paddle  wheel,  and  the  jet  and 
screw  propellers.  The  action  of  the  wind  upon  sails  will  not 
be  here  discussed,  as  lying  outside  of  the  scope  of  the  work. 

The  unit  of  measure  used  on  the  ocean  is  generally  the  nau- 
tical mile  or  knot,  which  is  about  6080  feet,  so  that  knots  per 
hour  may  be  transformed  into  feet  per  second  by  multiplying 
by  1.69,  and  feet  per  second  may  be  transformed  into  knots  per 
hour  by  multiplying  by  0.592.  On  rivers  the  statute  mile  is 
used,  and  the  corresponding  multipliers  will  be  1.47  and  0.682. 
On  the  ocean  the  weight  of  a  cubk  foot  of  water  is  to  be  taken 
as  about  64  pounds  (it  is  often  used  as  64.32  pounds,  so  that 
the  numerical  value  is  the  same  as  2g),  and  in  rivers  at  62.5 
pounds. 

The  speed  of  a  ship  at  sea  is  roughly  measured  by  observa- 
tions with  the  log,  which  is  a  triangular  piece  of  wood  attached 
to  a  cord  which  is  divided  by  tags  into  lengths  of  about  5o| 


366  NAVAL   HYDROMECHANICS.  [CiiAP.  XIIL 

feet.  The  log  being  thrown,  the  number  of  tags  run  out  in  half 
a  minute  is  the  same  as  the  number  of  knots  per  hour  at  which 
the  ship  is  moving,  since  5of  feet  is  the  same  part  of  a  knot  that 
a  half  minute  is  of  an  hour.  The  patent  log,  which  is  a  small 
self-recording  current  meter,  drawn  in  the  water  behind  the 
ship,  is  however  now  generally  used.  In  experimental  work 
more  accurate  methods  of  measuring  the  velocity  are  necessary, 
and  for  this  purpose  the  boat  may  run  between  buoys  whose 
distance  apart  has  been  found  by  triangulation  from  measured 
bases  on  shore. 

When  a  boat  or  ship  is  to  be  propelled  through  water,  the 
resistances  to  be  overcome  increase  with  its  velocity,  and  con- 
sequently,  as  in  railroad  trains,  a  practical  limit  of  speed  is  soon 
attained.  These  resistances  consist  of  three  kinds — the  dynamic 
pressure  caused  by  the  relative  velocity  of  the  boat  and  the 
water,  the  frictional  resistance  of  the  surface  of  the  boat,  and 
the  wave  resistance.  The  first  of  these  can  be  entirely  over- 
come, as  indicated  in  Art.  132,  by  giving  to  the  boat  a  "  fair" 
form,  that  is,  such  a  form  that  the  dynarqic  pressure  of  the  im- 
pulse near  the  bow  is  balanced  by  that  of  the  reaction  of  the 
water  as  it  closes  in  around  the  stern.  It  will'be  supposed  in 
the  following  pages  that  the  boat  has  this  form,  and  hence  this 
first  resistance  need  not  be  further  considered.  The  second 
and  third  sources  of  resistance  will  be  discussed  later. 

The  total  force  of  resistance  which  exists  when  a  vessel  is 
propelled  with  the  velocity  v  can  be  ascertained  by  drawing  it 
in  tow  at  the  same  velocity,  and  placing  on  the  tow  line  a  dy- 
namometer to  register  the  tension.  An  experiment  by  FROUDE 
on  the  Greyhound,  a  steamer  of  1157  tons,  gave  for  the  total 
resistance  the  following  figures  :  * 

At    4  knots  per  hour,       0.6  tons  ; 

At    6  knots  per  hour,        1.4  tons  ; 

*  THEARLE'S  Theoretical  Naval  Architecture,  London,  1876,  p.  347. 


ART.  148.]  FRICTIONAL  RESISTANCES.  367 

At  8  knots  per  hour,  2.5  tons; 
At  10  knots  per  hour,  4.7  tons  ; 
At  12  knots  per  hour,  9.0  tons. 

This  shows  that  at  low  speeds  the  resistance  varies  about  as 
the  square  of  the  velocity,  and  at  higher  speeds  in  a  faster 
ratio.  For  speeds  of  15  to  1 8  knots  per  hour — the  usual  ve- 
locity of  ocean  steamers — there  is  but  little  known  regarding 
the  resistance,  but  as  an  approximation  it  is  usually  taken  as 
varying  with  the  square  of  the  velocity. 

Prob.  173.  What  horse-power  was  expended  in  the  above 
test  of  the  Greyhound  when  the  speed  was  12  knots  per  hour? 

ARTICLE  148.  FRICTIONAL  RESISTANCES. 

When  a  stream  or  jet  moves  over  a  surface  its  velocity  is 
retarded  by  the  frictional  resistances,  or  if  the  velocity  be  main- 
tained uniform  a  constant  force  is  overcome.  In  pipes,  con- 
duits, and  channels  of  uniform  section  the  velocity  is  uniform, 
and  consequently  each  square  foot  of  the  surface  or  bed  exerts 
a  constant  resisting  force,  the  intensity  of  which  will  now  be 
approximately  computed.  This  resistance  will  be  the  same  as 
the  force  required  to  move  the  same  surface  in  still  water,  and 
hence  the  results  will  be  directly  applicable  to  the  propulsion 
of  ships. 

Let  F  be  the  force  of  frictional  resistance  per  square  foot 
of  surface  of  the  bed  of  a  channel,  p  its  wetted  perimeter,  /  its 
length,  h  its  fall  in  that  length,  a  the  area  of  its  cross-section, 
and  v  the  mean  velocity  of  flow.  The  force  of  friction  over 
the  entire  surface  then  is  Fpl,  and  the  work  per  second  lost  in 
friction  is  Fplv.  The  work  done  by  the  water  per  second  is 
Wh  or  wavh.  Equating  these  two  expressions  for  the  work, 

ah 

F  =  w  — -,  =  wrs. 
pi 


368  NAVAL  HYDROMECHANICS.  [CHAP.  XIII. 

Now,  inserting  for  rs  its  value  from  formula  (70)  of  Art.  94, 
there  results 


(98) 


in  which  w  is  the  weight  of  a  cubic  foot  of  water,  and  c  is  the 
coefficient  in  the  mean  velocity  formulas  whose  value  is  to  be 
taken  from  the  tables  in  Chapter  VIII.  Inasmuch  as  the  ve- 
locities along  the  bed  of  a  channel  are  somewhat  less  than  the 
mean  velocity  v,  the  values  of  F  thus  determined  will  probably 
be  slightly  greater  than  the  actual  resistance. 

For  smooth  iron  pipes  the  following  are  values  of  the  fric- 
tional  resistance  in  pounds  per  square  foot  of  surface  at  differ- 
ent velocities,  as  computed  from  the  above  formula : 

v    =        2.  4.  6.  10.  15. 

For  i  foot  diameter,  F  —  0.023,  0.080,  0.17,  0.43,  0.92; 
For  4  feet  diameter,  F  =  0.01$,  0.053,  o.ii,  °-28,  0.59. 

These  figures  indicate  that  the  resistance  is  subject  to  much 
variation  in  pipes  of  different  diameters ;  it  is  not  easy  to  con- 
clude from  them,  or  from  formula  (98),  what  the  force  of  re- 
sistance is  for  plane  surfaces  over  which  water  is  moving. 

Experiments  'made  by  moving  flat  plates  in  stilljyater  so 
that  the  direction  of  motion  coincides  with  the  plane  of  the 
surface  have  furnished  conclusions  regarding' the  laws  of  fluid 
friction  similar  to  those  deduced  from  the  flow  of  water  in  pipes. 
It  is  found  that  the  total  resistance  is  approximately  propor- 
tional to  the  area  of  the  surface,  and  approximately  propor- 
tional to  the  square  of  the  velocity.  Accordingly,  the  force  of 
resistance  per  square  foot  may  be  written 

F  =  fv\ (98)' 

in  which  f  is  a  number  depending  upon  the  nature  of  the  sur- 


ART.  148.]  FRICTIONAL  RESISTANCES.  369 

face.  The  following  are  average  values  of  f  for  large  surfaces, 
as  given  by  UNWIN  :  * 

Varnished  surface,  f  =  0.00250; 

Painted  and  planed  plank,  f  =  0.00339  ; 

Surface  of  iron  ships,  f  =  0.00351  ; 

Fine  sand  surface,  f  =  0.00405  ; 

New  well-painted  iron  plate,  f  =  0.00473. 

Undoubtedly  the  value  of  f  is  subject  to  variations  with  the 
velocity,  but  the  experiments  on  record  are  so  few  that  the  law 
and  extent  of  its  variation  cannot  be  formulated.  It  should, 
however,  be  remarked  that  the  formulas  and  constants  here 
given  do  not  apply  to  low  velocities,  for  the  reasons  given  in 
Art.  92.  At  the  same  time  they  are  only  approximately  ap- 
plicable to  high  velocities.  A  low  velocity  of  a  body  moving 
in  an  unlimited  stream  may  be  regarded  as  I  foot  per  second 
or  less,  a  high  velocity  as  25  or  30  feet  per  second. 

It  may  be  noted  that  the  above-mentioned  experiments  in- 
dicate that  the  value  of  F  is  greater  for  small  surfaces  than  for 
large  ones.  For  instance,  a  varnished  board  50  feet  long  gave 
f  =  0.00250,  while  one  20  feet  long  gave  f  •=.  0.00278,  and  one 
8  feet  long  gave  f  =  0.00325,  the  motion  being  in  all  cases  in 
the  direction  of  the  length.  The  resistance  is  the  same  what- 
ever be  the  depth  of  immersion,  for  the  friction  is  uninfluenced 
by  the  intensity  of  the  static  pressure.  This  is  proved  by  the 
circumstance  that  the  flow  of  water  in  a  pipe  is  found  to  de- 
pend only  upon  the  head  on  the  outlet  end,  and  not  upon  the 
pressure-heads  along  its  length. 

Prob.  174.  What  is  the  frictional  resistance  of  a  boat  when 
moving  at  the  rate  of  9  knots  per  hour,  the  area  of  its  immersed 
surface  being  320  square  feet,  and  f  =  0.0035  ? 

*  Encyclopaedia  Britannica,  gth  Edition,  vol.  xii.  p.  483. 


37°  NA  VAL   HYDROMECHANICS.  [CHAP.  XIIL 

ARTICLE  149.  WORK  REQUIRED  IN  PROPULSION. 

When  a  boat  or  ship  moves  through  still  water  with  a  ve- 
locity v,  it  must  overcome  the  pressure  due  to  impulse  of  the 
water  and  the  resistance  due  to  the  friction  of  its  surface  on  the 
water  and  air.  If  the  surface  be  properly  curved  there  is  no 
resultant  pressure  due  to  impulse,  as  shown  in  Art.  132.  The 
resistance  caused  by  friction  of  the  immersed  surface  on  the 
water  can  be  estimated,  as  explained  above.  If  A  be  the  area 
of  this  surface  in  square  feet,  the  work  per  second  required  to 
overcome  this  resistance  is 

k  =  AFv  =fAv* (99) 

The  work,  and  hence  the  horse-power,  required  to  move  a  boat 
accordingly  varies  approximately  as  the  cube  of  its  velocity. 
By  the  help  of  the  values  of /given  in  the  last  article  an  ap- 
proximate estimate  of  the  work  can  be  made  for  particular 
cases.  The  resistance  of  the  air,  which  in  practice  must  be 
considered,  will  be  here  neglected. 

To  illustrate  this  law  let  it  be  required  to  find  how  many 
tons  of  coal  will  be  used  by  a  steamer  in  making  a  trip  of  3000 
miles  in  6  days,  when  it  is  known,  that  800  tons  are  used  in 
making  the  trip  in  10  days.  As  the  power  used  is  proportional 
to  the  amount  of  coal,  and  as  the  distances  travelled  per  day  in 
the  two  cases  are  500  miles  and  300  miles,  the  law  gives 

T  5s 


,  a  ' 


6  X  80        3; 

whence  T  —  2220  tons.  By  the  increased  speed  the  expense 
for  fuel  is  increased  277  per  cent,  while  the  time  is  reduced  40 
per  cent.  If  the  value  of  wages,  maintenance,  interest,  etc., 
saved  on  account  of  the  reduction  in  time,  will  balance  the 
extra  expense  for  fuel,  the  increased  speed  is  profitable.  That 
such  a  compensation  occurs  in  many  instances  is  apparent  from 


ART.  150.]  THE  JET  PROPELLER.  371 

the  constant  efforts  to  reduce  the  time  of  trips  of  passenger 
steamers. 

When  a  boat  moves  with  the  velocity  v  in  a  current  which 
has  a  velocity  u  in  the  same  direction  the  velocity  of  the  boat 
relative  to  the  water  is  v  —  u,  and  the  resistance  is  proportional 
to  (v  —  uj  and  the  work  to  (v  —  u)*.  If  the  boat  moves  in  the 
opposite  direction  to  the  current  the  relative  velocity  is  v  -\-  u, 
and  of  course  v  must  be  greater  than  u  or  no  progress  would 
be  made.  In  all  cases  of  the  application  of  the  formulas  of 
this  article  and  the  last,  v  is  to  be  taken  as  the  velocity  of  the 
boat  relative  to  the  water. 

Another  source  of  resistance  to  the  motion  of  boats  and 
ships  is  the  production  of  waves.  This  is  due  in  part  to  a 
different  level  of  the  water  surface  along  the  sides  of  the  ship 
due  to  the  variation  in  static  pressure  caused  by  the  velocity, 
and  in  part  to  other  causes.  It  is  plain  that  waves,  eddies,  and 
foam  cause  energy  to  be  dissipated  in  heat,  and  that  thus  a 
portion  of  the  work  furnished  by  the  engines  of  the  boat  is  lost. 
This  source  of  loss  is  supposed  to  consume  from  10  to  40  per 
cent  of  the  total  work,  and  it  is  known  to  increase  with  the 
velocity.  On  account  of  the  uncertainty  regarding  this  resist- 
ance, as  well  as  those  due  to  the  friction  of  the  water  and  air, 
practical  computations  on  the  power  required  to  move  boats  at 
given  velocities  can  only  be  expected  to  furnish  approximate 
results. 

Prob.  175.  Compute  the  horse-power  required  for  a  ve- 
locity of  1 8  knots  per  hour,  taking  A  —  7473  square  feet  and 
/  =  0.004. 

ARTICLE  150.  THE  JET  PROPELLER. 

The  method  of  jet  propulsion  consists  in  allowing  water  to 
enter  the  boat  and  acquire  its  velocity,  and  then  to  eject  it 
backwards  at  the  stern  by  means  of  a  pump.  The  reaction  thus 


372  NA  VAL  HYDROMECHANICS.  [CHAP.  XIII. 

produced  propels  the  boat  forward.  To  investigate  the  effi- 
ciency of  this  method,  let  W  be  the  weight  of  water  ejected 
per  second,  V  its  velocity  relative  to  the  boat,  and  v  the  ve- 
locity of  the  boat  itself.  The  absolute  velocity  of  the  issuing 
water  is  then  V  —  v,  and  it  is  plain  without  further  discussion 
that  the  maximum  efficiency  will  be  obtained  when  this  is  o, 
or  when  V  =  v,  as  then  there  will  be  no  energy  remaining 
in  the  water  which  is  propelled  backward.  It  is,  however,  to 
be  shown  that  this  condition  can  never  be  realized. 

The  work  which  is  lost  in  the  absolute  velocity  of  the 
water  is 

W 
k'=-(V-V}\ 

The  work  which  is  exerted  on  the  boat  by  the  reaction  is 

W 

k=  —  (V-  v)v. 

o 

The  sum  of  these  is  the  total  theoretic  work,  or 

W 

K=-(V>-v*). 

Therefore  the  efficiency  of  jet  propulsion  is  expressed  by 

k  zv 


This  becomes  equal  to  unity  when  v  —  V  as  before  indicated, 
but  then  it  is  seen  that  the  work  k  becomes  o  unless  Wis  infinite. 
The  value  of  Wis  waV,  if  a  be  the  area  of  the  orifices  through 
which  the  water  is  ejected  ;  and  hence  in  order  to  make  e  unity 
and  at  the  same  time  perform  work  it  is  necessary  that  either 
V  or  a  should  be  infinity.  The  jet  propeller  is  therefore  like 
a  reaction  wheel  (Art.  143),  and  it  is  seen  upon  comparison  that 
the  formula  for  efficiency  is  the  same  in  the  two  cases. 

By  equating  the  above  value  of  the  useful  work  to  that 
established  in  the  last  article  there  is  found 
fgAv*=  waV(V  -  v)\ 


ART.  151.]  PADDLE    WHEELS.  373 

and  if  this  be  solved  for  F,  and  the  resulting  value  be  substi- 
tuted in  (100),  it  reduces  to 

4 


e  = 


3  + 


wa 


which  again  shows  that  e  approaches  unity  as  the  ratio  of  a  to 
A  increases.  The  area  of  the  orifices  of  discharge  must  hence 
be  very  large  in  order  to  realize  both  high  power  and  high 
efficiency.  For  this  reason  attempts  to  propel  vessels  by  this 
method  have  not  proved  practically  successful.  In  nature  the 
same  result  is  seen,  for  no  marine  animal  except  the  cuttle-fish 
uses  this  principle  of  prqpulsion.  Even  the  cuttle-fish  cannot 
depend  upon  his  jet  to  escape  from  his  enemies,  but  for  this 
relies  upon  his  supply  of  ink,  with  which  he  darkens  the  water 
about  him. 

Prob.  176.  Compute  the  approximate  area  of  the  orifices 
for  a  jet  propeller  to  run  at  10  knots  per  hour  when  exerting 
1 200  horse-power,  the  efficiency  being  0.67. 

ARTICLE  151.  PADDLE  WHEELS. 

The  method  of  propulsion  by  rowing  and  paddling  is  famil- 
iar to  all.  The  power  is  furnished  by  muscular  energy  within 
the  boat,  the  water  is  the  fulcrum  upon  which  the  blade  of  the 
oar  acts,  and  the  force  of  reaction  thus  produced  is  transmitted 
to  the  boat  and  urges  it  forward.  If  water  were  an  unyielding 
substance,  the  theoretic  efficiency  of  the  oar  should  be  unity, 
or,  as  in  any  lever,  the  work  done  by  the  force  at  the  rowlock 
should  equal  the  work  performed  by  the  motive  force  exerted. 
But  as  the  water  is  yielding,  some  of  it  is  driven  backward  by 
the  blade  of  the  oar,  and  thus  energy  is  lost. 

The  paddle  or  side  wheel  so  extensively  used  in  river  navi- 
gation is  similar  in  principle  to  the  oar.  The  former  is  furnished 


374  NA  VAL   HYDROMECHANICS.  [CHAP.  XIII. 

by  a  motor  within  the  boat,  the  blades  or  vanes  of  the  wheel 
tend  to  drive  the  water  backward,  and  the  reaction  thus  pro- 
duced urges  the  boat  forward.  On  first  thought  it  might  be 
supposed  that  the  efficiency  of  the  method  would  be  governed 
by  laws  similar  to  those  of  the  undershot  wheel,  and  such  would 
be  the  case  if  the  vessel  were  stationary  and  the  wheel  were 
used  as  an  apparatus  for  moving  the  water.  In  fact,  however, 
the  theoretic  efficiency  of  the  paddle  wheel  is  much  higher  than 
that  of  the  undershot  motor. 

The  work  exerted  by  the  steam-engine  upon  the  paddle 
wheels  may  be  represented  by  PV,  in  which  P  is  the  pressure 
produced  by  the  vanes  upon  the  water,  and  Fis  their  velocity 
of  revolution  ;  and  the  work  actually  imparted  to  the  boat  may 
be  represented  by  Pv,  in  which  v  is  its  velocity.  Accordingly 
the  efficiency  of  the  paddle,  neglecting  losses  due  to  foam  and 

waves,  is 

v  _       v 

€  ^^  -fj.  —         j         , 
K          V  -f-  V 

in  which  vl  is  the  difference  F—  v,  or  the  so-called  "  slip."  If 
the  slip  be  o,  the  velocities  Fand  v  are  equal,  and  the  theoretic 
efficiency  is  unity.  The  value  of  F  is  determined  from  the 
radius  r  of  the  wheel  and  its  number  of  revolutions  per  minute ; 
thus  V •=•  2nrN. 

On  account  of  the  lack  of  experimental  data  it  is  difficult  to 
give  information  regarding  the  practical  efficiency  of  paddle 
wheels  considered  from  a  hydromechanic  point  of  view.  Ow- 
ing to  the  water  which  is  lifted  by  the  blades,  and  to  the  foam 
and  waves  produced,  much  energy  is  lost.  They  are,  however, 
very  advantageous  on  account  of  the  readiness  with  which  the 
boat  can  be  stopped  and  reversed.  When  the  wheels  are  driven 
by  separate  engines,  as  is  sometimes  done  on  river  boats,  per- 
fect control  is  secured,  as  they  can  be  revolved  in  opposite 
directions  when  desired.  Paddle  wheels  with  feathering  blades 


ART.  152.]  THE   SCREW  PROPELLER.  375 

are  more  efficient  than  those  with  fixed  radial  ones,  but  prac- 
tically they  are  found  to  be  cumbersome,  and  liable  to  get  out 
of  order.*  In  ocean  navigation  the  screw  has  now  almost  en- 
tirely replaced  the  paddle  wheel  on  account  of  its  higher 
efficiency. 

Prob.   177.  Ascertain  the  size  of  the  paddle  wheels  of  the 
steamship  Great  Eastern. 


ARTICLE  152.  THE  SCREW  PROPELLER. 

The  screw  propeller  consists  of  several  helicoidal  blades 
attached  at  the  stern  of  a  vessel  to  the  end  of  a  horizontal 
shaft  which  is  made  to  revolve  by  steam  power.  The  dynamic 
pressure  of  the  reaction  developed  between  the  water  and  the 
helicoidal  surface  drives  the  vessel  forward,  the  theoretic  work 
of  the  screw  being  the  product  of  this  pressure  by  the  distance 
traversed.  The  pitch  of  the  screw  is  the  distance,  parallel  to 
the  shaft,  between  any  point  on  a  helix,  and  the  corresponding 
point  on  the  same  helix  after  one  turn  around  the  axis,  and 
the  pitch  may  be  constant  at  all  distances  from  the  axis,  or  it 
may  be  variable.  If  the  water  were  unyielding,  the  vessel 
would  advance  a  distance  equal  to  the  pitch  at  each  revolution 
of  the  shaft  ;  actually,  the  advance  is  less  than  the  pitch,  the 
difference  being  called  the  slip.  The  effect  thus  is  that  the 
pressure  P  existing  between  the  helical  surfaces  and  the  water 
moves  the  vessel  with  the  velocity  #,  while  the  theoretic  velocity 
which  should  occur  is  F,  being  the  pitch  of  the  screw  multi- 
plied by  the  number  of  revolutions  per  second.  The  work 
expended  is  hence  PV  or  P(v  -f-  vt),  if  vl  be  the  slip  per  second, 
and  the  work  utilized  is  Pv.  Accordingly  the  efficiency  of  screw 
propulsion  is,  approximately, 

v 

s?    — —     

*  For  description  of  these,  see  KNIGHT'S  Mechanical  Dictionary. 


376  NAVAL   HYDROMECHANICS.  [CHAP.  XIIL 

which  is  the  same  expression  as  before  found  for  the  paddle 
wheel.  Here,  as  in  the  last  article,  all  the  pressure  exerted 
by  the  blades  upon  the  water  is  supposed  to  act  backward  in  a 
direction  parallel  to  the  shaft  of  the  screw,  and  the  above 
conclusion  is  approximate  because  this  is  actually  not  the  case, 
and  also  because  the  action  of  friction  has  not  been  considered. 

The  pressure  P  which  is  exerted  by  the  helicoidal  blades 
upon  the  water  is  the  same  as  the  thrust  or  stress  in  the  shaft, 
and  the  value  of  this  may  be  approximately  ascertained  by  re- 
garding it  as  due  to  the  reaction  of  a  stream  of  water  of  cross- 
section  a  and  velocity  ^,  or 


Another  expression  for  this  may  be  found  from  the  expended 
work  k  ;  thus  : 

/»=*. 

», 

Numerical  values  computed  from  these  two  expressions  do 
not,  however,  agree  well,  the  latter  giving  in  general  a  much 
less  value  than  the  former. 

In  Art.  149  the  work  to  be  performed  in  propelling  a  vessel 
of  fair  form  whose  submerged  surface  is  A  was  found  to  be 


If  the  value  of  v  is  taken  from  this  and  inserted  in  the  ex 
pression  for  efficiency,  there  obtains 


e  = 


which  shows  that  e  increases  as  vl ,  f,  and  A  decrease,  and  as 
£  increases.  Or  for  given  values  of  /and  A  the  efficiency  de- 
creases with  the  speed. 


ART.  152.]  THE   SCREW  PROPELLER.  377 

It  has  been  observed  in  a  few  instances  that  the  slip  vl  is 
negative,  or  that  V,  as  computed  from  the  number  of  revolu- 
tions and  pitch  of  the  screw,  is  less  than  v.  This  is  probably 
due  to  the  circumstance  that  the  water  around  the  stern  is 
following  the  vessel  with  a  velocity  v't  so  that  the  real  slip  is 
V  —  v  +  v1  instead  of  V ••-  v.  The  existence  of  negative  slip 
is  usually  regarded  as  evidence  of  poor  design. 

In  some  cases  twin  screws  are  used,  as  with  these  the  vessel 
can  be  more  readily  controlled.  Fig.  107  shows  the  twin 
screws  of  the  City  of  New  York,  an  ocean  steamer  of  580  feet 


FIG  107. 

length,  63.5  feet  breadth,  and  42  feet  depth,  with  a  gross  ton- 
nage of  10  500  and  an  estimated  horse-power  of  about  16000. 
These  are  made  to  revolve  in  opposite  directions.  The  usual 
practice,  however,  is  to  have  but  one  screw.  The  practical 
advantage  of  the  screw  over  the  paddle  wheel  has  been  found 
to  be  very  great,  and  this  is  probably  due  to  the  circumstance* 
that  less  energy  is  wasted  in  lifting  the  water  and  in  forming 
waves. 

Prob.  178.  Ascertain  the  size  and  the  pitch  of  the  screws 
on  the  steamer  City  of  New  York.  Compute  the  theoretic 
efficiency  if  the  number  of  revolutions  per  minute  is  150  when 
the  velocity  of  the  steamer  is  20  knots  per  hour. 


3/8 


NAVAL  HYDROMECHANICS. 


[CHAP.  XIII. 


ARTICLE  153.  THE  ACTION  OF  THE  RUDDER. 

The  action  of  the  rudder  in  steering  a  vessel  involves  a 
principle  that  deserves  discussion.    In  Fig.  108  is  shown  a  plan 

of  a  boat  with  the  rudder  turned  to 
the   starboard   side,  at   an   angle   9 
-C-  with  the  line  of  the  keel.     The  ve- 
locity of  the  vessel  being  v,  the  ac- 
tion of  the  water  upon  the  rudder  is 
the  same  as  if  the  vessel  were  at  rest 
FIG.  108.  and  the  water  in  motion  with  the 

velocity  v.     Let  Wbe  the  weight  of  water  which  produces  dy- 

W 
namic  pressure  against  the  rudder,  due  to  the  impulse  —  v  (Art. 

<?> 
128).     The  component  of  this  pressure  normal  to  the  rudder  is 

P=—  vsinO, 
g 

and  its  effect  in  turning  the  vessel  about  the  centre  of  gravity 
C  is  measured  by  its  moment  with  reference  to  that  point. 
Let  b  be  the  breadth  of  the  rudder,  and  d  the  distance  CH  be- 
tween the  centre  of  gravity  and  the  hinge  of  the  rudder  ;  then 
the  lever  arm  of  the  force  P  is 


'  cos  0, 
and  accordingly  the  turning  moment  is 

W 

M  —  —  v(b  sin  0  +  d  s 
2g  v 


20). 


To  determine  that  value  of  6  which  produces  the  greatest  effect 
in  turning  the  boat  the  derivative  of  M  with  respect  to  8  must 
vanish,  which  gives 

r  /  70 

C05(>=-^  +  V12+6W 


ART.  154.]  TIDES  AND    WA  VES.  379 

and  from  this  the  value  of  6  is  found  to  be  approximately  45°, 
since  d  is  always  much  larger  than  b. 

The   following  are   values  of  0  for  several  values  of  the 
ratio  b  -±-  d\ 

**•**'*  t  TO  T^IT  O 

cos  0  =  0.6825       0.6916       0.6947       0.7069       0.7071 
#  =  46°  58'      46°  1 5'      46°  oo'      45°oi/        45° 

In  practice  it  is  usual  to  arrange  the  mechanism  of  the  rudder 
so  that  it  can  only  be  turned  to  an  angle  of  about  42°  with  the 
keel,  for  it  is  found  that  the  power  required  to  turn  it  the  addi- 
tional 3°  or  4°  is  not  sufficiently  compensated  by  the  slightly 
greater  moment  that  would  be  produced.  The  reasoning  also 
shows  that  intensity  of  the  turning  moment  increases  with  v,  so 
that  the  rudder  acts  most  promptly  when  the  boat  is  moving 
rapidly.  For  the  same  reason  a  rudder  on  a  steamer  propelled 
by  a  screw  does  not  need  to  be  so  broad  as  on  one  driven  by 
paddle  wheels,  for  the  effect  of  the  screw  is  to  increase  the  ve- 
locity of  the  impinging  water,  and  hence  also  its  dynamic  pres- 
sure against  the  rudder. 

Prob.  179.  Explain  how  it  is  that  a  ship  can  sail  against  the 
wind. 

ARTICLE  154.  TIDES  AND  WAVES. 

The  complete  discussion  of  the  subject  of  waves  might,  like 
so  many  other  branches  of  Hydraulics,  be  expanded  so  as  to 
embrace  an  entire  treatise,  and  hence  there  can  be  here  given 
only  the  briefest  outline  of  a  few  of  the  most  important  prin- 
ciples. There  are  two  classes  or  kinds  of  waves,  the  first  in- 
cluding the  tidal  waves  and  those  produced  by  earthquakes  or 
other  sudden  disturbances,  and  the  second  those  due  to  the 
wind.  The  daily  tidal  wave  generated  by  the  attraction  of  the 
moon  and  sun  originates  in  the  South  Pacific  Ocean,  whence 


38o 


NA  VAL  H  YDR OME CHA NICS. 


[CHAP.  XIIL 


it  travels  in  all  directions  with  a  velocity  dependent  upon  the 
depth  of  water  and  the  configuration  of  the  continents,  and 
which  in  some  regions  is  as  high  as  1000  miles  per  hour. 
Striking  against  the  coasts,  the  tidal  waves  cause  currents  in 
inlets  and  harbors,  and  if  the  circumstances  were  such  that 
their  motion  could  become  uniform  and  permanent,  these  might 
be  governed  by  the  same  laws  which  apply  to  the  flow  of  water 
in  channels.  Such,  however,  is  rarely  the  case  ;  and  according- 
ly the  subject  of  tidal  currents  is  one  of  much  complexity,  and 
not  capable  of  general  formulation. 

The  velocity  of  a  wave  produced  by  a  sudden  disturbance 
in  a  channel  of  uniform  width  is  found  by  experiments,  and 
also  by  theoretic  considerations,  to  be  VgD,  where  D  desig- 
nates the  depth  of  water.  When  such  a  wave  advances  into 
shallow  water  its  height  is  observed  to  increase,  and  when  D 
becomes  as  small  as  one-half  the  height  of  the  wave  it  breaks 
into  foam. 

Rolling  waves  produced  by  the  wind  travel  with  a  velocity 
which  is  small  compared  with  those  of  the  first  class,  although 
in  water  where  the  disturbance  can  extend  to  the  bottom  it  is 
generally  supposed  that  their  speed  is  also  represented  by 


FIG.  109. 


Upon  the  ocean  the  maximum  length  of  such  waves  is 
estimated  at  550  feet,  and  their  velocity  at  about  53  feet  per 
second.  For  this  class  of  waves  it  is  found  by  observation  that 
each  particle  of  water  upon  the  surface  moves  in  an  elliptic  or 
circular  orbit,  whose  time  of  revolution  is  the  same  as  the  time 


ART.  154.]  TIDES  AND    WAVES.  381 

of  one  wave  length.  Thus  the  particles  on  the  crest  of  a  wave 
are  moving  forward  in  the  direction  of  the  motion  of  the  wave, 
while  those  in  the  trough  are  moving  backward.  When  such 
waves  advance  into  shallow  water  their  length  and  speed  de- 
crease, but  the  time  of  revolution  of  the  particles  in  their  orbits 
remains  unaltered,  and  as  a  consequence  the  slopes  become 
steeper  and  the  height  greater,  until  finally  the  front  slope  be- 
comes vertical,  and  the  wave  breaks  with  roar  and  foam.  Be- 
low the  surface  the  particles  revolve  also  in  elliptic  orbits,  which 
grow  smaller  in  size  toward  the  bottom.  The  curve  formed 
by  the  vertical  section  of  the  surface  of  a  wave  at  right  angles 
to  its  length  is  of  a  cycloidal  nature. 

The  force  exerted  by  ocean  waves  when  breaking  against 
sea  walls  is  very  great,  as  already  mentioned  in  Art.  132,  and 
often  proves  destructive.  If  walls  can  be  built  so  that  the 
waves  are  reflected  without  breaking,  as  is  sometimes  possible 
in  deep  water,  their  action  is  rendered  less  injurious.  Upon 
the  ocean  waves  move  in  the  same  direction  as  the  wind,  but 
along  shore  it  is  observed  that  they  move  normally  toward  it, 
whatever  may  be  the  direction  in  which  the  wind  is  blowing. 

Prob.  1 80.  In  a  channel  6.5  feet  wide,  and  of  a  depth  de- 
creasing uniformly  1.5  feet  per  1000,  BAZIN  generated  a  wave 
by  suddenly  admitting  water  at  the  upper  end.  At  points 
where  the  depths  were  2.16,  1.85,  1.46,  and  0.80  feet,  the  ve- 
locities were  observed  to  be  8.70,  8.67,  7.80,  and  6.69  feet  per 
second.  Do  these  velocities  agree  with  the  law  above  stated  ? 


APPENDIX. 


APPENDIX. 


ANSWERS  TO  PROBLEMS. 

Below  will  be  found  the  answers  to  most  of  the  problems; 
whose  solution  is  not  stated  in  the  text,  the  number  of  the 
problem  being  enclosed  in  parenthesis.  A  few  answers  have 
been  purposely  omitted  in  order  that  the  student  may  be 
thrown  entirely  upon  his  own  resources.  However  satisfac- 
tory it  may  be  to  know  in  advance  the  result  of  the  solution 
of  an  exercise,  let  the  student  bear  in  mind  that  after  com- 
mencement day  answers  to  problems  will  not  be  given  him. 

Chapter  I.  (i),  996,  or,  in  round  numbers,  1000  kilos  per 
square  centimeter.  (3),  393  pounds,  47.1  gallons,  178  kilos. 
(5)>  73-8  pounds,  5.02  atmospheres.  (9),  14.73  pounds.  (10), 
7.85  gallons,  65.6  pounds. 

Chapter  II.  (12),  100  feet,  100  meters.  (16),  5280  pounds. 
(17),  2880  feet.  (18),  8.04  feet,  2000  pounds.  (19),  y  —  \d. 
(22),  8.7  feet  and  8.4  feet.  (23),  8.7  pounds. 

Chapter  IIL  (27),  v  =  8.75  and  2.19  feet  per  second. 
(29),  0.0534  cubic  feet  per  second.  (33),  15.3  and  19.6  cubic 
feet  per  second.  (34),  90.1  feet  per  second.  (35),  94.6  feet 
per  second.  (37),  f  \' 2  times  that  for  the  hemisphere.  (39), 
318.  (42),  1. 8 1  horse-powers. 


ANSWERS   TO  PROBLEMS.  383 

Chapter  IV.  (47),  24  feet.  (48),  0.617.  (49),  0.981.  (50), 
11.97  (52),  0.601.  (53),  0.0169  cubic  feet  per  second.  (54), 
0.605.  (56),  17.3  feet.  (58),  101.  (60),  6.05  cubic  feet  per 
second.  (61),  0.88.  (62),  0.18  of  one  per  cent.  (64),  4  min- 
utes, 13  seconds. 

Chapter  V.  (66),  2  feet  per  second.  (68),  0.00173  feet. 
(69),  0.837  feet  per  second,  and  0.0109  feet.  (70),  1.451,  1.458, 
1.465  cubic  feet  per  second.  (73),  4.035  cubic  feet  per  second. 
(74),  7.10  and  6.97  cubic  feet  per  second.  (75),  21.1  cubic  feet 
per  second.  (76),  1.82  feet.  (77),  7.58  feet.  (79),  6.6  X  3.3 
feet,  8.53  X  1-5  feet. 

Chapter  VI.  (80),  5.69  and  5.46  horse-powers.  (81),  0.985. 
(82),  0.962.  (84),  —  2.89  pounds,  6.67  feet.  (85),  0.802.  (89), 
0.28.  feet.  (90,0.995. 

Chapter  VII.  (92),  0.135.  (93),  0.29  feet.  (94),  7.64  and 
8.44  feet.  (96),  3.39  feet  per  second.  (98),  6.14  gallons  per 
minute.  (100),  3.06  and  4.80  inches.  (101),  26  gallons  per 
minute.  (103),  0.935  and  0.722  feet.  (104),  32.  (105),  6.13 
cubic  feet  per  second.  (108),  velocity-head  =  9.34  feet,  (no), 
2.75  feet  per  second,  68  feet.  (112),  15.9  horse-powers.  (114), 
15.8  cubic  feet  per  second,  0.913  and  0.705  feet.  (115),  56440. 

Chapter  VIII.  (116),  i  foot.  (117),  2.54  feet  per  second. 
(119),  226.5  cubic  feet  per  second.  (120),  4.4  feet.  (121),  1.214 
feet  and  7.13  feet  per  second.  (122),  I  :  1.2 1.  (124),  5.20  and 
3.69  feet  persecond.  (125),  6 1  300000.  (126),  64  750000.  (127), 
57630000.  (131),  0.8 1  horse-powers. 

Chapter  IX.  (133),  547  cubic  feet  per  second.  (134),  364 
pounds.  (135),  4.85  feet  persecond.  (137),  0.81  feet  per  sec- 


3^4  APPENDIX. 

ond.     (139),   1.59  feet  per  second.     (142),  770  cubic  feet  per 
second.     (145),  4.5  feet  at  one  mile. 

Chapter  X.  (147),  0.00266.  (149),  0.32.  (150),  0.488  horse- 
powers, and  35.4  per  cent  efficiency. 

Chapter  XL  (154),  3.97.  (155),  32.1  pounds.  (158),  93.2 
pounds.  (160),  34.5  feet  per  second.  (161),  507.  (163),  0.82. 

Chapter  XII.  (164),  1.04.  (165),  49.2.  (166),  27.2.  (172), 
see  descriptions  of  methods  of  arranging  the  Jonval  turbine. 

Chapter  XIII.  (173),  576.  (174),  195  pounds.  (175),  1000. 
076)>  53-7  square  feet.  (177),  56  feet  in  diameter.  (180),  com- 
puted velocities  are  8.34,  7.71,  6.86,  and  5.11  feet  per  second. 


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